Unit 3: Solving Quadratic Equations and Their Applications and Simplifying Compound Fractions In this unit, you will study complex numbers and the arithmetic of complex numbers. Remember from the last unit, you cannot take the square root of a negative number and get a real number. Instead, you will get an imaginary number. Now you can study using the radical to solve equations that contain a power. After that, you will be introduced to quadratic equations and different techniques used to solve quadratic equations. Quadratic equations are equations that have x^{2} as an expression. You will also study applications involving quadratic equations. One of the primary applications is distance above the ground equation due to gravity. If you throw an object in the air, you know that the ball will come back to the ground due to gravity. The distance that the object is above the ground at any time is a quadratic equation. Solving radical equations and simplifying compound fractions are also studied.
Unit 3 Time Advisory
Completing this unit should take approximately 30 hours.
☐ Subunit 3.1: 4.25 hours
☐ Subunit 3.2: 5 hours
☐ Subunit 3.3: 4 hours
☐ Subunit 3.4: 4.5 hours
☐ Subunit 3.5: 4 hours
☐ Subunit 3.6: 4.25 hours
☐ Subunit 3.7: 4 hours
Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- multiply and divide complex numbers;
- simplify expressions with complex roots;
- solve quadratic equations by completing the square and using the
quadratic formula;
- solve equations with rational exponents;
- solve equations by taking roots;
- solve radical equations with even and odd roots;
- solve simple application problems involving quadratic equations; and
- simplify compound fractions.
3.1 Complex Numbers
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 8, Section 8.8: Complex Numbers”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 8, Section 8.8: Complex
Numbers”
(PDF)
Instructions: Read Section 8.8 in Chapter 8 of your textbook, pages
318–324, to learn about complex numbers. Complex numbers are a super
set of the real numbers. They consist of a real part and an
imaginary part, which is a real number times the square root of -1
(usually denoted by i). Note that this reading also covers the
topics in Subunits 3.1.1–3.1.5.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be
found here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Complex Numbers”
Link: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Complex
Numbers”
(PDF)
Instructions: Complete pages 52–56 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.1.1–3.1.5, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.1.1 Simplifying Square Roots of Negatives Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Square Roots of Negatives”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Square Roots of
Negatives” (YouTube)
Instructions: Watch the video linked above, which defines imaginary numbers, or complex numbers. It is necessary to understand complex numbers when solving quadratic equations since the square root of a negative number is common. Note that the definition of √-1 = i and i^{2} = -1.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.1.2 Adding/Subtracting Complex Numbers Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Add/Subtract”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Add/Subtract”
(YouTube)
Instructions: Watch the video linked above, which discusses adding and subtracting complex numbers. Basically, the rules for adding and subtracting complex numbers are the same as adding and subtracting like terms in an algebraic expression if you view i as an unknown.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.1.3 Multiplying Complex Numbers Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Multiply”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Multiply”
(YouTube)
Instructions: Watch the video linked above, which discusses multiplying complex numbers. Multiplication of complex numbers follows the same rules as multiplying any algebraic expression. Just remember that i^{2} should be replaced by -1.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.1.4 Rationalizing Monomials with Complex Numbers Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Rationalize Monomials”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Rationalize
Monomials” (YouTube)
Instructions: Watch the video linked above, which discusses rationalizing monomials with complex numbers. Thus, when you have a fraction with an i in the denominator, to remove the i from the denominator, multiply the numerator and the denominator by i. The denominator will have i^{2}, which is -1.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.1.5 Rationalizing Binomials with Complex Numbers Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Rationalize Binomials”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complex
Numbers – Rationalize
Binomials” (YouTube)
Instructions: Watch the video linked above, which discusses rationalizing fractions with binomial terms of complex numbers. Recall i^{2} = -1 and (a + bx)(a - bx) = a^{2} - b^{2}x^{2}. The conjugate of a + bi is a - bi. Multiplying the numerator and the denominator of a rational expression by the conjugate of the denominator will remove the i from the denominator.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2 Equations with Radicals
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.1: Solving with
Radicals”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.1: Solving with
Radicals” (PDF)
Instructions: Read Section 9.1 in Chapter 9 of your textbook, pages
326–331, to learn about solving equations with radicals. When
solving these equations, you will want to do the opposite of a
radical, raising each side to a power which is the same as the
index. Be sure that the radical is isolated (has no other terms on
that side of the equation). Note that this reading also covers the
topics in Subunits 3.2.1–3.2.8.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Equations with Radicals”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab Notebook: “Equations with
Radicals”
(PDF)
Instructions: Complete pages 57–60 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.2.1–3.2.8, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.2.1 Solving Equations with Radicals That Have Odd Roots Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Odd Roots”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Odd Roots”
(YouTube)
Instructions: Watch the video linked above, which discusses solving equations with radicals that have odd indices, such as the cube root, fifth root, etc. Raising the equation (both sides) to the odd power should reduce the equation to one that you know how to solve.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.2 Solving Equations with Radicals That Have Even Roots (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Even Roots (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Even Roots (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses equations involving radicals with even indices, such as square root, fourth root, etc. This case is different from the odd roots. You must take an additional step and check your answers since it is common to pick up extraneous solutions.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.3 Solving Equations with Radicals That Have Even Roots (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Even Roots (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Even Roots (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of an equation involving radicals with even roots.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.4 Isolating Roots (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Isolate Roots (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Isolate Roots (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses solving equations involving radicals with another term on the same side of the equation. It is very important to be sure that the term containing a radical is on one side of the equation alone. In other words, you must isolate the radical to one side of the equation.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.5 Isolating Roots (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Isolate Roots (Part 2)” (YouTube)
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Isolate Roots (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of isolating the radical before raising each side to a power.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.6 Equations with Two Roots (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Two Roots (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Two Roots (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses solving equations involving two radicals. When there are multiple radicals in an equation with other terms, solving the equation becomes increasingly more complicated because you cannot isolate each radical. Even when you raise each side of the equation to a power, you will still have a radical on one side. Isolating this radical to one side and raising both sides to the power of this index will remove the radical.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.7 Equations with Two Roots (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Two Roots (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Two Roots (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of two radicals in an equation.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.8 Equations with Two Roots (Part 3) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Radicals – Two Roots Part 3”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Radicals – Two Roots Part
3” (YouTube)
Instructions: Watch the video linked above, which presents yet another example of two radicals in an equation.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.3 Equations with Exponents
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.2: Solving with
Exponents”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.2: Solving with
Exponents”
(PDF)
Instructions: Read Section 9.2 in Chapter 9 of your textbook, pages
332–336, to learn how to solve equations with exponents using
radicals. Powers (algebraic expressions with exponents) and radicals
are opposites, or inverses. This is similar to subtraction being the
inverse of addition and division being the inverse of
multiplication. So it is natural to use radicals to solve equations
with powers. Note that this reading also covers all the topics in
Subunits 3.3.1–3.3.4.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Equations with Exponents”
Link: Tyler Wallace’s Intermediate Algebra Lab
Notebook: “Equations with
Exponents”
(PDF)
Instructions: Complete pages 62–65 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.3.1–3.3.4, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.3.1 Equations with Odd Exponents Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Exponents – Odd Exponents”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Exponents – Odd
Exponents” (PDF)
Instructions: Watch the video linked above, which discusses solving equations with odd exponents. For instance, suppose x^{3} =- Then, if you take the cube root of each side you get x = 3.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
- Then, if you take the cube root of each side you get x = 3.
3.3.2 Equations with Even Exponents Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Exponents – Even Exponents”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Exponents – Even
Exponents” (YouTube)
Instructions: Watch the video linked above, which discusses solving equations with even exponents. This case is slightly different than odd powers because you need to place a ± in front of the answer. For instance, x^{2} = 4 then x = ±2. You can check this answer by substituting x = +2 for x and then x = -2 for x. You will see that both +2 and -2 are solutions.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.3.3 Isolating the Exponent Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Exponents – Isolate Exponent”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Exponents – Isolate
Exponent” (PDF)
Instructions: Watch the video linked above, which discusses solving equations with exponents in which you must first isolate the term with the exponent. This example parallels the case where a radical must be isolated before proceeding.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.3.4 Rational Exponents Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Equations
with Exponents – Rational Exponents”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Equations with
Exponents – Rational
Exponents” (YouTube)
Instructions: Watch the video linked above, which discusses solving equations with rational exponents. Now that you have studied solving equations with radicals and solving equations with powers, you can combine the two into rational exponents.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.4 Completing the Square
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.3: Complete the
Square”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.3: Complete the
Square”
(PDF)
Instructions: Read section 9.3 in Chapter 9 of your textbook, pages
337–342, to learn how to solve a quadratic equation by completing
the square. Quadratic equations are equations with a square term in
the equation. If there is no linear term, then you can solve the
equation by using the technique of solving equations with even
exponents studied in Subunit 3.3.2. If the equation has both a
linear term and a constant term, there is only one method that works
for all quadratic equations. This subunit discusses the technique of
reformulating the equation so that there is a perfect square on one
side. Then you can use techniques you’ve already learned to solve
this equation. Note that this reading covers the topics in Subunits
3.4.1–3.4.3.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Complete the Square”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab Notebook: “Complete the
Square”
(PDF)
Instructions: Complete pages 66–68 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.4.1–3.4.3, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.4.1 Finding the Constant to Complete the Square Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.4.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complete
the Square – Find c”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complete the
Square – Find c”
(YouTube)
Instructions: Watch the video linked above. In the expression x^{2} + bx, you want to add a c to this expression so that it is a perfect square. If you add (b/2)^{2}, then
x^{2} + bx + (b/2)^{2} = (x+b/2)^{2}
And c = (b/2)^{2}.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.4.2 Rational Solutions when Completing the Square Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.4.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complete
the Square – Rational Solutions”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complete the
Square – Rational
Solutions” (YouTube)
Instructions: Watch the video linked above, which discusses the completing the square technique for solving quadratic equations in which the solutions are rational solutions – a ratio of two integers.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.4.3 Irrational and Complex Solutions when Completing the Square Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.4.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Complete
the Square – Irrational and Complex Solutions”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Complete the
Square – Irrational and Complex
Solutions” (YouTube)
Instructions: Watch the video linked above, which discusses the completing the square technique for solving quadratic equations in which the square root is not a perfect square and/or is a negative number. If the square root is not a perfect square, then the solutions will have radicals in them. If the square root is a negative, then the solutions will be complex numbers.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.5 The Quadratic Formula
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.4: Quadratic Formula”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.4: Quadratic
Formula”
(PDF)
Instructions: Read Section 9.4 in Chapter 9 of your textbook, pages
343–347, to learn how to solve quadratic equations using the
quadratic formula. The quadratic formula for solving a quadratic
equation is the result of solving a general quadratic equation in
the form ax^{2} + bx + c = 0 by using the completion of the
square technique. Note that this reading covers the topics in
Subunits 3.5.1–3.5.4.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Quadratic Formula”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab Notebook: “Quadratic
Formula”
(PDF)
Instructions: Complete pages 69–72 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.5.1–3.5.4, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.5.1 Deriving the Quadratic Formula Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic
Formula – Finding the Formula”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic
Formula – Finding the
Formula” (YouTube)
Instructions: Watch the video linked above, which discusses how to use the completion of the square technique with a general quadratic equation to develop the quadratic formula. The general quadratic equation in standard form is ax^{2} + bx + c = 0. Using techniques in this unit for completing the square will result in the quadratic formula.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.5.2 Using the Quadratic Formula Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic
Formula – Using the Formula”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic
Formula – Using the
Formula” (YouTube)
Instructions: Watch the video linked above, which illustrates how to use the quadratic formula. You must always be sure that the quadratic equation is in the standard form (equal to 0). Then, carefully plug the corresponding a, b, and c into the formula and simplify all arithmetic expressions.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.5.3 Rearranging Terms in a Quadratic Equation to Make Equal to Zero Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.5.
Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic Formula – Make Equal to Zero” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic Formula – Make Equal to Zero” (YouTube)
Instructions: Watch the video linked above, which presents another example of using the quadratic formula. In this case the video illustrates with an example that is not in the standard form, ax^{2} + bx + c = 0, and how to put it into this form prior to using the quadratic formula.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.5.4 The Quadratic Formula with No Linear Term Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic
Formula – No Linear Term (b=0)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Quadratic
Formula – No Linear Term
(b=0)” (YouTube)
Instructions: Watch the video linked above, which discusses using the quadratic formula with no linear term. This video shows how to apply the quadratic formula when b = 0. Remember, however, that if b = 0, you can also apply the techniques learned in Subunit 3.4.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.6 Rectangles
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.7: Application:
Rectangles”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.7: Application:
Rectangles”
(PDF)
Instructions: Read Section 9.7 in Chapter 9 of your textbook, pages
357–363, to learn an application of the quadratic equation:
rectangles. There are several examples, but basically you know from
geometry that the area of a rectangle is A = LW where L is the
length and W is the width of the rectangle. If L and W are related
(say you know the L of your rectangle is 3 units longer than the W)
then the A is equal to some quadratic relationship of L or W. If you
know the area of your rectangle, you now have a quadratic equation
to determine L or W. Note that this reading covers the topics in
Subunits 3.6.1–3.6.5.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Rectangles”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab
Notebook: “Rectangles”
(PDF)
Instructions: Complete pages 73–75 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.6.1–3.6.5, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.6.1 Using Quadratic Equations Given the Area of a Rectangle Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.6.
Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles – Area” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles – Area” (YouTube)
Instructions: Watch the video linked above, which discusses an application of the quadratic equations, rectangles, given a specific area. Suppose you need to build a dog pen of 350 square feet and, according to the size, you know the length must be twice as much as the width – 3 feet – what size pen should you construct?
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.6.2 Using Quadratic Equations Given the Perimeter of a Rectangle (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.6.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles
– Perimeter (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles –
Perimeter (Part 1)”
(YouTube)
Instructions: Watch the video linked above, which discusses the rectangle application of quadratic equations given the perimeter. Instead of explicitly giving a relationship between the length and the width, you are given the perimeter of the rectangle. From the perimeter and the formula for the perimeter, P = 2L + 2W, you can create a relationship between L and W. As in the previous example, you can then create a quadratic equation given the area of the rectangle.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.6.3 Using Quadratic Equations Given the Perimeter of a Rectangle (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.6.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles
– Perimeter (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles –
Perimeter (Part 2)”
(YouTube)
Instructions: Watch the video linked above, which presents another example of the rectangle application of quadratic equations given the perimeter.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.6.4 Making a Rectangle Bigger (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.6.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Make It
Bigger (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Make It Bigger
(Part 1)” (YouTube)
Instructions: Watch the video linked above, which discusses the rectangle and what effects increasing each side have on the area. As an example, suppose a square is increased by 6 inches. This results in having an area of 16 times the original square. What is the dimension of the original square?
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.6.5 Making a Rectangle Bigger (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.6.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles
– Make It Bigger (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rectangles –
Make It Bigger (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of what happens when increasing the lengths of the sides of a rectangle.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.7 Compound Fractions
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 7, Section 7.5: Complex Fractions”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 7, Section 7.5: Complex
Fractions”
(PDF)
Instructions: Read Section 7.5 in Chapter 7 of your textbook, pages
262–267, to learn how to simplify and work with compound fractions.
Compound fractions are fractions in which the numerator is a
fraction, the denominator is a fraction, or both are fractions. Note
that this reading also covers the topics in Subunits 3.7.1–3.7.4.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Compound Fractions” Link: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Compound Fractions” (PDF)
Instructions: Complete pages 76–79 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 3.7.1–3.7.4, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
3.7.1 Compound Fractions with Numbers (No Unknowns) Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.7.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Numbers”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Numbers”
(YouTube)
Instructions: Watch the video linked above, which discusses how to simplify and handle compound fractions. If you multiply the numerator and denominator by the lowest common denominator (LCD) of all of the denominators, then you will have a single normal fraction.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.7.2 Compound Fractions with Monomial Denominators Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.7.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Monomials”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Monomials”
(YouTube)
Instructions: Watch the video linked above, which discusses how to simplify and work with compound fractions with monomials – single algebraic equations – in the denominators.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.7.3 Compound Fractions with Binomials in the Denominators Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.7.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Binomials”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Binomials”
(YouTube)
Instructions: Watch the video linked above, which discusses how to simplify and work with compound fractions with binomial expressions in the denominators of each fraction.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
3.7.4 Compound Fractions with Negative Exponents Note: This subunit is also covered by the reading and assessment assigned in Subunit 3.7.
- Lecture: YouTube: Tyler “Compound Fractions – Negative
Exponents”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Compound
Fractions – Negative
Exponents” (YouTube)
Instructions: Watch the video linked above, which discusses how to simplify and work with negative exponents. Recall that a negative power is the same as the reciprocal of the expression. So, a fraction that has terms with a negative exponent is a compound fraction. This video works through an example with negative exponents in both the numerator and the denominator.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.