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QUADRATIC EQUATIONS AND APPLICATIONS
Copyright © Cengage Learning. All rights reserved.
What You Should Learn
• Solve quadratic equations by factoring.
• Solve quadratic equations by extracting square
roots.
• Solve quadratic equations by completing the
square.
• Use the Quadratic Formula to solve quadratic
equations.
• Use quadratic equations to model and solve
real-life problems.
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Factoring
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Factoring
A quadratic equation in x is an equation that can be
written in the general form
ax2 + bx + c = 0
where a, b, and c are real numbers with a  0. A quadratic
equation in x is also called a second-degree polynomial
equation in x.
In this section, you will study four methods for solving
quadratic equations: factoring, extracting square roots,
completing the square, and the Quadratic Formula.
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Factoring
The first method is based on the Zero-Factor Property.
If ab = 0, then a = 0 or b = 0.
Zero-Factor Property
To use this property, write the left side of the general form
of a quadratic equation as the product of two linear factors.
Then find the solutions of the quadratic equation by setting
each linear factor equal to zero.
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Example 1(a) – Solving a Quadratic Equation by Factoring
2x2 + 9x + 7 = 3
Original equation
2x2 + 9x + 4 = 0
Write in general form.
(2x + 1)(x + 4) = 0
2x + 1 = 0
x+4=0
The solutions are x =
original equation.
Factor.
x=
Set 1st factor equal to 0.
x = –4
Set 2nd factor equal to 0.
and x = –4. Check these in the
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Example 1(b) – Solving a Quadratic Equation by Factoring
6x2 – 3x = 0
Original equation
3x(2x – 1) = 0
3x = 0
2x – 1 = 0
cont’d
Factor.
x=0
Set 1st factor equal to 0.
x=
Set 2nd factor equal to 0.
The solutions are x = 0 and x = . Check these in the
original equation.
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Factoring
Be sure you see that the Zero-Factor Property works only
for equations written in general form (in which the right side
of the equation is zero).
So, all terms must be collected on one side before
factoring.
For instance, in the equation (x – 5)(x + 2) = 8, it is
incorrect to set each factor equal to 8.
To solve this equation, you must multiply the binomials on
the left side of the equation, and then subtract 8 from each
side.
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Factoring
After simplifying the left side of the equation, you can use
the Zero-Factor Property to solve the equation.
Try to solve this equation correctly.
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Extracting Square Roots
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Extracting Square Roots
Solving an equation of the form u2 = d without going
through the steps of factoring is called extracting square
roots.
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Example 2 – Extracting Square Roots
Solve each equation by extracting square roots.
a. 4x2 = 12
b. (x – 3)2 = 7
Solution:
a. 4x2 = 12
x2 = 3
x=
Write original equation.
Divide each side by 4.
Extract square roots.
When you take the square root of a variable expression,
you must account for both positive and negative
solutions.
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Example 2 – Solution
So, the solutions are x =
in the original equation.
b. (x – 3)2 = 7
x–3=
x=3
The solutions are x = 3 
equation.
cont’d
and x = –
. Check these
Write original equation.
Extract square roots.
Add 3 to each side.
. Check these in the original
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Completing the Square
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Completing the Square
Note that when you complete the square to solve a
quadratic equation, you are just rewriting the equation so it
can be solved by extracting square roots.
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Example 3 – Completing the Square: Leading Coefficient Is 1
Solve x2 + 2x – 6 = 0 by completing the square.
Solution:
x2 + 2x – 6 = 0
Write original equation.
x2 + 2x = 6
Add 6 to each side.
x2 + 2x + 12 = 6 + 12
Add 12 to each side.
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Example 3 – Solution
(x + 1)2 = 7
x+1=
x = –1 
The solutions are x = –1 
equation as follows.
cont’d
Simplify.
Take square root of each side.
Subtract 1 from each side.
. Check these in the original
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Example 3 – Solution
cont’d
Check:
x2 + 2x – 6 = 0
(–1 +
)2 + 2 (–1 +
8–2
–2+2
)–6≟0
–6≟0
8–2–6=0
Write original equation.
Substitute –1 +
for x.
Multiply.
Solution checks.
Check the second solution in the original equation.
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Completing the Square
When solving quadratic equations by completing the
square, you must add (b/2)2 to each side in order to
maintain equality.
If the leading coefficient is not 1, you must divide each side
of the equation by the leading coefficient before completing
the square.
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The Quadratic Formula
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The Quadratic Formula
The Quadratic Formula is one of the most important
formulas in algebra. You should learn the verbal statement
of the Quadratic Formula:
“Negative b, plus or minus the square root of b
squared minus 4ac, all divided by 2a.”
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The Quadratic Formula
In the Quadratic Formula, the quantity under the radical
sign, b2 – 4ac, is called the discriminant of the quadratic
expression ax2 + bx + c. It can be used to determine the
nature of the solutions of a quadratic equation.
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The Quadratic Formula
If the discriminant of a quadratic equation is negative, as in
case 3 above, then its square root is imaginary (not a real
number) and the Quadratic Formula yields two complex
solutions.
When using the Quadratic Formula, remember that before
the formula can be applied, you must first write the
quadratic equation in general form.
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Example 6 – The Quadratic Formula: Two Distinct Solutions
Use the Quadratic Formula to solve x2 + 3x = 9.
Solution:
The general form of the equation is x2 + 3x – 9 = 0. The
discriminant is b2 – 4ac = 9 + 36 = 45, which is positive. So,
the equation has two real solutions.
You can solve the equation as follows.
x2 + 3x – 9 = 0
Write in general form.
Quadratic Formula
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Example 6 – Solution
cont’d
Substitute a = 1, b = 3,
and c = –9.
Simplify.
Simplify.
The two solutions are:
Check these in the original equation.
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The Quadratic Formula: One Solution
Use the Quadratic Formula to solve 8x2 – 24x + 18 = 0.
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