Unit 4 Topic 3: Solving
Quadratic Equations
Algebra 1 Summit
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2014 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: August 12, 2014
AUTHOR
Algebra 1 Summit
www.ck12.org
C HAPTER
Chapter 1. Unit 4 Topic 3: Solving Quadratic Equations
1
Unit 4 Topic 3: Solving
Quadratic Equations
Solving Quadratic Equations
There are multiple ways to solve a quadratic equation. Not every method works for every equation, so it is important
to be fluent in multiple methods. Some methods will be easier than others depending on the format of the equation.
The solutions to a quadratic equation are when the function’s y values are equal to zero. A quadratic function can
have one, two or no real solutions.
Method 1: The Table Method
This method is as simple as looking at the table of values for the solutions. Input the equation into the graphing
calculator in y= then select 2nd Graph to find the table of values. The solutions will be the x values which correspond
to the y values of zero.
Example 1: x2 + x − 6 = 0
TABLE 1.1:
x
-4
-3
-2
-1
0
1
2
3
y
6
0
-4
-6
-6
-4
0
6
The solutions of this equation are x=-3 and x=2, because they are the x values, which correspond with the y value of
zero.
Although this method seems very simple, it does not work for every example.
Example 2: x2 + 3x − 5 = 0
This is the table of values for this function.
TABLE 1.2:
x
-5
-4
-3
-2
-1
0
1
y
5
-1
-5
-7
-7
-5
-1
1
www.ck12.org
Method 2: The Graphing Method
The solutions of this equation are not visible in this table and therefore this method cannot be used. It can be
estimated that one solution is between -4 and -5 and the other is between 1 and 2, but another method must be used
to find the exact solutions.
This method is as simple as looking at the graph of the function to find the solutions. The solutions will be the x
values which correspond to the y values of zero. Look to the x axis for the solutions.
FIGURE 1.1
http://jodzilla.wikispaces.com/.
The solutions of this equation are x = -1 and x = 3 because these are the x values which correspond with the y value
of zero. The solutions are also called the “zeros” for this reason.
Example 2: −x2 + 2x + 2 = 0
Although this method seems very simple, it does not work for every example without a calculator.
FIGURE 1.2
http://www.mathamazement.com/Lessons/Pre-Calculus/02_Polynomial-and-Rational-Functions/quadratic-functions.h
tml
The exact solutions of this equation are not visible on this graph and therefore this method cannot be used. It
can be estimated that one solution is between -1 and 0 and the other solution is between 2 and 3, but another
method must be used to find the exact solutions.
In order to solve this using a graph, use of the graphing calculator is required.
2
www.ck12.org
Chapter 1. Unit 4 Topic 3: Solving Quadratic Equations
Move curser near these locations.
To find the solutions, hit 2nd Trace.
Select 5: intersect
Move the curser along the curve towards one of the zeros. When the curser is near the height y = 0, hit enter 3 times.
Repeat these steps for the other side.
The solutions are x = -0.732 and x = 2.732
Method 3: The Zero Product Property
The Zero Product Property states that for any factors a · b = 0, then either a = 0, b = 0, or they are both zero.
Video on Factoring and ZPP:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%201:%20Solving%20a%20
quadratic%20equation%20by%20factoring
If a quadratic equation can be written in factored form, this method can be used. Since not all quadratic functions
can be factored, this method will not work on every problem.
Example 1: (x + 3)(x – 5) = 0
Since this example includes two factors that are multiplied to make zero, take each factor and set them equal to zero
to solve.
x + 3 = 0 and x – 5 = 0
x = -3 and x = 5
The solutions are -3 and 5.
To check the solutions, try each solution in the original equation.
(-3 + 3)(-3 – 5) = 0 and (5 + 3)(5 – 5) = 0
The values of -3 and 5 both satisfy the equation.
Example 2: x2 − 4x − 21 = 0
In this example, the problem must first be factored to be in the form a · b = 0.
(x + 3)(x – 7) = 0
Now that this example includes two factors that are multiplied to make zero, take each factor and set them equal to
zero to solve.
TABLE 1.3:
x+3=0
x = -3
x-7=0
x=7
The solutions are -3 and 7.
To check the solutions, try each solution in the original equation.
3
www.ck12.org
TABLE 1.4:
(-3)2 – 4(-3) – 21 = 0
0=0
(7)2 – 4(7) – 21 = 0
0=0
The values of -3 and 7 both satisfy the equation.
Example 3: x2 + 2x + 2 = 0
In this example, the quadratic must first be factored to be in the form a · b = 0. Since this quadratic function cannot
be factored, this method cannot be used.
Method 4: The Square Root Method
The Square Root Method is most often used when a quadratic function is written in vertex form a(x – h)2+ k = 0.
The equation must be manipulated into vertex form by completing the square if it is not given in this form.
The first step is to have the perfect square alone on one side of the equation. Eliminate the “k” value by adding
25 to the other side.
(x – 2)2 – 25 = 0
Next, take the square root of both sides to cancel the square with the square root. This leaves only (x – 2) on
the left.
(x – 2)2 = 25
There are two numbers that when squared result in 25. Set the quantity (x-2) equal to these two values, 5 and
-5. Solve each equation for x to find the two solutions.
TABLE 1.5:
(x – 2) = 5
X -2 +2 = 5+ 2
x=7
(x – 2) = -5
x - 2 + 2 = -5 + 2
x=-3
Solve each equation for x to find the two solutions.
x = 7 and x = -3
Example 2: −2(x − 3)2 + 12 = 0
The first step is to have the perfect square alone on one side of the equation. Eliminate the “k” value by subtracting
12 to the other side.
-2(x – 3)2 +12 = 0
-2(x – 3)2 = -12
Divide each side by -2 to isolate the quantity (x – 2)2 on the left.
√
√
(x – 3)2 = 6
Next, take the square root of both sides to cancel the square. This leaves only (x -3) on the left and
√
(x-3) = 6
There are two numbers that when squared results in 6. Since the number is irrational, leave the value as
√
6
Check the solutions by evaluating both values in the original equation. Both solutions should result as a true
4
www.ck12.org
Chapter 1. Unit 4 Topic 3: Solving Quadratic Equations
FIGURE 1.3
statement. Since these values were rounded, the check may be off by a small margin.
Example 3: x2 − 2x − 5 = 0
This example is not given in vertex form. In order to use this method, the equation must be re-written in vertex form
by completing the square. For a review on completing the square, watch the video below.
Completing the Square Video:
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%201:%20Solving%20a%20quadratic%
FIGURE 1.4
Check the solutions by evaluating both values in the original equation. Both solutions should result as a true
statement. Since these values were rounded, the check may be off by a small margin.
5
www.ck12.org
Method 5: The Quadratic Formula
The Quadratic Formula can be used to solve quadratic functions for their solutions. The formula works because it
is simply the formula of a quadratic function in standard form (ax2 + bx + c = 0) solved for x. Students do not have
to derive the formula in Algebra 1, but will need to know how it is used.
p
−b±
(b2 ) − 4ac
The Quadratic Formula:
2a
Step 1: Make sure the problem is solved to the form ax2 + bx + c = 0.
Step 2: Identify the values of a, b and c.
Step 3: Plug in the values of a, b and c into the formula.
Step 4: Simplify using order of operations and your calculator.
Example 1:x2 − 2x − 15 = 0
The solutions are 5 and -3.
Check the solutions in the original equation.
6
www.ck12.org
Chapter 1. Unit 4 Topic 3: Solving Quadratic Equations
Example 2: 2x2 + 6x = −4
FIGURE 1.5
Check the solutions in the original equation.
2(-1)2 + 6(-1) = -4 and 2(-2)2 + 6(-2) = -4
-4 = -4 and -4 = -4
7
www.ck12.org
Example 3: x2– 2x + 15 = 0
Hint!
The value under the square root symbol in the formula is called the discriminant. This number identifies how many
solutions the equation will have. Use the chart below for reference:
TABLE 1.6:
Discriminant Value
Positive Number
Zero
Negative Number
Number of Solutions
Two Solutions
One Solution
Zero Real Solutions
Examples 1 and 2 had a positive number discriminant value and resulted in two solutions. Example 3 had a negative
number discriminant value and resulted in zero real solutions. Example 4 below is an example of a discriminant
value of zero which will result in only one solution value.
Example 4: x2 + 16 = 8x
Step 1: This problem must be solved into standard form. Subtract the 8x to the left side and place it as the “b” value.
x2 – 8x + 16 = 0
Step 2: a = 1, b = -8, c = 16The solution is 4.
Check the solution in the original equation.
(4)2 – 8(4) + 16 = 0
0=0
Quadratic Formula Video:
https://www.khanacademy.org/math/algebra/quadratics/quadratic-formula/v/quadratic-formula-1
8
www.ck12.org
Chapter 1. Unit 4 Topic 3: Solving Quadratic Equations
The next two examples will show how to solve an equation using all of the methods discussed in this chapter.
FIGURE 1.6
9
www.ck12.org
FIGURE 1.7
10