Download Solving Quadratic Equations Using the Square Root Property

Quadratic Equations,
Quadratic Functions, and
Complex Numbers
Copyright © Cengage Learning. All rights reserved.
9
Section
9.1
Solving Quadratic Equations
Using the Square Root Property
Copyright © Cengage Learning. All rights reserved.
Objectives
1. Use the zero-factor property to solve a
1
quadratic equation.
22. Use the square root property to solve a
quadratic equation.
33. Solve an application involving a quadratic
equation.
3
1.
Use the zero-factor property to solve a
quadratic equation
4
Use the zero-factor property to solve a quadratic equation
We know that the definition of a quadratic equation.
A quadratic equation in one variable is an equation of the
form
ax2 + bx + c = 0 (a  0)
where a, b, and c are real numbers.
We have solved quadratic equations by factoring and using
the zero-factor property. If ab = 0, then a = 0 or b = 0.
5
Use the zero-factor property to solve a quadratic equation
The zero-factor property states that when the product of
two numbers is 0, at least one of the numbers is 0.
For example, the equation (x – 4)(x + 5) = 0 indicates that a
product is equal to 0. By the zero-factor property, one of
the factors must be 0:
x–4=0
or
x+5=0
We can solve each of these linear equations to get
x=4
or
x = –5
The equation (x – 4)(x + 5) = 0 has two solutions: 4 and –5.
6
Use the zero-factor property to solve a quadratic equation
Factoring Method
To solve a quadratic equation by factoring, we
1. Write the equation in ax2 + bx + c = 0 form (called
quadratic or standard form).
2. Factor the left side of the equation.
3. Use the zero-factor property to set each factor equal to 0.
4. Solve each resulting linear equation.
5. Check each solution in the original equation.
7
Example
Solve: 6x2 – 3x = 0.
Solution:
Since the equation is already in quadratic form, we begin
by factoring the left side of the equation.
6x2 – 3x = 0
3x(2x – 1) = 0
Factor out 3x, the GCF.
By the zero-factor property, we have
3x = 0
or
2x – 1 = 0
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Example – Solution
cont’d
We can solve these linear equations to get
3x = 0
or
2x – 1 = 0
x=0
2x = 1
Check:
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Example – Solution
The solutions are 0 and
cont’d
.
10
Use the zero-factor property to solve a quadratic equation
The factoring method doesn’t work with many quadratic
equations.
For example, the trinomial in the equation
x2 + 5x + 1 = 0 cannot be factored by using rational
coefficients.
To solve such equations, we need to develop other
methods. The first of these methods uses the square root
property.
11
2.
Use the square root property to solve
a quadratic equation
12
Use the square root property to solve a quadratic equation
If x2 = c, then x is a number whose square is c. Since
and
, the equation x2 = c has two
solutions.
Square-Root Property
The equation x2 = c has two solutions:
or
We can write the result with double-sign notation. The
equation
(read as “x equals plus or minus
means that
or
.
”)
13
Example
Solve: x2 = 16.
Solution:
The equation x2 = 16 has two solutions:
or
Using double-sign notation, we have
.
14
Example – Solution
cont’d
Check:
The solutions are 4 and –4.
15
3.
Solve an application involving a
quadratic equation
16
Example – Integers
The product of the first and third of three consecutive odd
integers is 77. Find the integers.
Analyze the problem
Some examples of three consecutive odd integers are
3, 5, and 7
and
–29 , –27, and –25
These examples illustrate that to obtain the next
consecutive odd integer from the previous one, we must
add 2.
17
Example – Integer Problems
cont’d
If we let x represent the first of three consecutive odd
integers, then x + 2 represents the second odd integer and
x + 2 + 2 or x + 4 represents the third odd integer.
Form an equation
Since the product of the first integer and the third integer is
x(x + 4) and we are given that this product is 77, we have
the equation
x(x + 4) = 77
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Example – Integer Problems
cont’d
Solve the equation
We can solve the equation as follows:
x(x + 4) = 77
x2 + 4x – 77 = 0
(x + 11)(x – 7) = 0
x + 11 = 0
x = –11
or
Apply the distributive property and subtract 77
from both sides.
Factor.
x–7=0
Set each factor equal to 0.
x=7
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Example – Integer Problems
cont’d
State the conclusion
Since the first consecutive odd integer x can be either
7 or –11, there are two possible sets of integers. If the first
integer is 7, the second is 9, and the third is 11. If the first
integer is –11, the second is –9, and the third is –7.
Check the results
If the three consecutive odd integers are 7, 9, and 11, the
product of the first and third integers is 7(11) = 77.
20
Example – Integer Problems
cont’d
If the three consecutive odd integers are –11, –9, and –7,
the product of the first and third integers is –11(–7) = 77.
The answers check and the integers are 7, 9, and 11 or
–11, –9, and –7.
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