Chapter 11
Solve Quadratic
Equations by Factoring
Solving Quadratic Equations and Inequalities
Tip
Is there a faster way to find the possible
solutions for a quadratic equation?
Method 1
Once you have honed your factoring skills, they
can be put to good use solving quadratic equations.
A quadratic equation is an equation in which the
highest exponent is 2 and is generally simplified
down to the form ax 2 + bx + c = 0. Sometimes,
however, you will run into an equation in which
c is equal to 0. These equations have the form of
ax 2 + bx = 0 and are simple to solve by factoring.
For more information on factoring, see chapter 10.
One method of solving by factoring requires you to
start by factoring the variable out of the equation
Step 1: Factor the Variable Out of the Equation
so the equation takes the form of x(ax + b) = 0.
You should then separate each factor into an
equation that equals zero, because if two factors
multiply to equal zero, then at least one of the
factors must equal zero. You will get one possible
solution for x right off the bat: x = 0. Solve for
the variable in the second equation to get the
second possible solution for x.
If an equation is in the form ax2 + bx = 0,
where a and b are numbers, the two
possible solutions will always be x = 0 and
x = – ba . For example, for the equation
2x2 – 10x = 0 below, the two solutions are
x = 0 and x = –( –10
2 ) = 5—the same result
that we obtained below. Using this trick
to find the solutions can help you quickly
solve quadratic equations.
ctice
Pra
Solve the following equations by factoring.
You can check your answers on page 263.
1) x 2 – 7 x = 0
2) x 2 + 5 x = 0
3) 3 x 2 – 8 x = 0
4) 2 x 2 – 20 x = 0
5) 3 x 2 + 3 x = 2 x – x 2
6) 2 x 2 – 4 x + 5 = x 2 + 5
You can check the possible solutions you found
by plugging your answers into the original
equation to see if they both satisfy the equation.
Step 2: Write Each Factor as Its Own Equation
Step 3: Find the Second Solution
Step 4: Check Your Answer
x
2 x 2 – 10 x
2(0) 2 – 10(0)
0–0
0
Let
2 x – 10 = 0
x (2 x – 10) = 0
2
2 x – 10 x = 0
2 x – 10 + 10 = 0 + 10
2 x = 10
x (2 x – 10) = 0
x =0
• You can use this method to
solve a quadratic equation
when the equation has
only two terms in the form
ax 2 + bx = 0 , where a and
b are numbers.
1 Factor the left side of the
equation by factoring out
the variable, such as x.
206
Note: When you factor
an equation, you break
the equation into pieces,
called factors, which
multiply together to
produce the original
equation. To factor a
variable out of an
equation, see page 188.
2 Write each factor as its
own equation and set
each factor equal to 0.
2 x = 10
2
2
x=5
2 x – 10 = 0
• You now have one
solution to the problem.
In this example, x can
equal 0.
Note: x = 0 will always be
one solution to the problem.
3 To determine the second
solution to the problem,
determine which numbers
you need to add, subtract,
multiply and/or divide to
place the variable by itself
on the left side of the
equation.
4 Add, subtract, multiply
• In this example, add 10 to
both sides of the equation.
Then divide both sides of
the equation by 2.
Note: For more information on
adding, subtracting, multiplying
and dividing numbers in
equations, see pages 64 to 67.
• You have found the second
and/or divide both sides of
solution to the original
the equation by the numbers equation. In this example,
you determined in step 3.
x can equal 5 .
x
2 x 2 – 10 x
2(5) 2 – 10(5)
2(25) – 50
50 – 50
0
Let
=0
=0
=0
=0
= 0 Correct!
=5
=0
=0
=0
=0
= 0 Correct!
5 To check your answer, place
the numbers you found into
the original equation and
solve the problems. If both
sides of the equation are
equal in both cases, you
have correctly solved the
equation.
• In this example,
the solutions to
the equation are
0 and 5 .
CONTINUED
207
Chapter 11
Solve Quadratic
Equations by Factoring continued
Solving Quadratic Equations and Inequalities
Ti p
Method 2
As you have seen, quadratic equations are generally
simplified down to the form ax 2 + bx + c = 0. While
there are different ways of solving different types
of quadratic equations, factoring is a foolproof
method you can use to solve all types.
To begin using the method of solving by factoring
shown below, set one side of the quadratic
equation to equal zero by moving all the terms to
the left side of the equation. Then you can factor
the equation. Once you have the equation down to
two factors, you can split the equation into two
Step 1: Set One Side of the Equation to Zero
equations with one side of each equation set to
equal zero. With the two smaller equations you
now have, solving for x is straightforward.
In some cases, you may end up with two factors
that are identical, such as (x + 2)(x + 2) = 0. This
leads to two identical solutions, which is called a
double root.
After factoring an equation, I ended up
with a factor that is a number. What
should I do?
You can ignore any factor that is a
number because these factors are so easy
to eliminate. Just divide both sides of the
equation by the number and it disappears.
For example, when you factor the equation
4x 2 + 28x + 40 = 0, you end up with the
equation 4(x + 2)(x + 5) = 0. Dividing both
sides by 4 leaves you with (x + 2)(x + 5) = 0.
Again, it is important to check that you have
not made an error in your calculations. Plug your
answers into the original equation to see if they
are correct.
Step 2: Factor the Problem
Step 3: Solve Each Equation
ctice
Pra
Solve for the variable by factoring
in the following equations. You can
check your answers on page 263.
1) x 2 – x – 2 = 0
2) 2 x 2 – 5 x – 3 = 0
3) 6 x 2 – 19 x + 10 = 0
4) x 2 + 8 x – 10 = 7 x + 10
5) 2 x 2 + 5 x = –2 x 2 + 8 x + 1
6) x 2 – 2 x + 2 = 2 x – 2
Step 4: Check Your Answer
Let
x 2 + 7 x = –10
x +2=0
x 2 + 7 x + 10 = 0
x 2 + 7 x = –10
x = –2
x +2–2=0–2
(–2) 2 + 7(–2) = –10
4 + (–14) = –10
x = –2
( x + 2)( x + 5) = 0
–10 = –10 Correct!
x 2 + 7 x + 10 = –10 + 10
x 2 + 7 x + 10 = 0
Let
x +2=0
x +5=0
x +5=0
x = –5
x 2 + 7 x = –10
x +5–5=0–5
x = –5
(–5) 2 + 7(–5) = –10
25 + (–35) = –10
–10 = –10 Correct!
• You can use this
method to solve any
quadratic equation.
1 Determine which
numbers and variables
you need to add and/or
subtract to place all the
numbers and variables
on the left side of the
equation and make
the right side of the
equation equal to 0.
208
2 Add and/or subtract the
numbers and variables
you determined in step 1
on both sides of the
equation.
• In this example, add 10
to both sides of the
equation.
Note: For more information
on adding and subtracting
numbers and variables in
equations, see page 64.
3 Factor the left side of the
equation.
Note: When you factor a
problem, you break the problem
into pieces, called factors, which
multiply together to produce the
original problem. You can use
the techniques you learned in
chapter 10 to factor a problem.
In this example, see page 198
to factor the problem.
4 Write each factor
as its own equation
and set each factor
equal to 0 .
5 In each equation,
determine which
numbers you need to
add, subtract, multiply
and/or divide to place the
variable by itself on the
left side of the equation.
6 Add, subtract, multiply
and/or divide both sides
of the equation by the
numbers you determined
in step 5.
• In this example,
subtract 2 from both
sides of the first
equation. Subtract 5
from both sides of
the second equation.
• Each answer is a
solution to the
original equation.
7 To check your answer,
place the numbers you
found into the original
equation and solve the
problems. If both sides
of the equation are
equal in both cases,
you have correctly
solved the equation.
• In this example,
the solutions to
the equation are
–2 and – 5.
209