Download 8.1

Chapter 8
Quadratic
Functions
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-1
1
Chapter Sections
8.1 – Solving Quadratic Equations by
Completing the Square
8.2 – Solving Quadratic Equations by the
Quadratic Formulas
8.3 – Quadratic Equations: Applications and
Problem Solving
8.4 – Writing Equations in Quadratic Form
8.5 – Graphing Quadratic Functions
8.6 – Quadratic and Other Inequalities in One
Variable
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-2
2
§ 8.1
Solving Quadratic
Equations by
Completing the Square
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-3
3
Quadratic Equations
A quadratic equation is an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers and a  0.
In Section 5.8 we solved quadratic equations by
factoring. In this section we introduce two
additional procedures used to solve quadratic
equations: the square root property and completing
the square.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-4
4
Square Root Property
Square Root Property
If x2 = a, where a is a real number, then x = ± √a.
Solve the equation x2 = 49.
x  49
2
x   49
x  7
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-5
5
Square Root Property
Solve the equation x2 - 9 = 0.
2
x 9  0
x2  9
x 9
x  3
Solve the equation x2 + 10 = 85.
x  10  85
2
x 2  75
x   75   25 5  5 3
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-6
6
Understand Perfect Square Trinomials
A perfect square trinomial is a trinomial that can
be expressed as the square of a binomial.
x2 – 10x + 25 = (x – 5) (x – 5) = (x – 5)2
a2 + 8a + 16 = (a + 4) (a + 4) = (a + 4)2
p2 – 14p + 49 = (p – 7) (p – 7) = (p – 7)2
Note that in every perfect square trinomial, the
constant term is the square of one-half the coefficient of
the x-term.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-7
7
Completing the Square
To Solve a Quadratic Equation by Completing the Square
1. Use the multiplication (or division) property of equality, if
necessary, to make the leading coefficient 1.
2. Rewrite the equation with the constant by itself on the right side
of the equation.
3. Take one-half the numerical coefficient of the first-degree term,
square it, and add this quantity to both sides of the equation.
4. Factor the trinomial as the square of a binomial.
5. Use the square root property to take the square root of both
sides of the equation.
6. Solve for the variable.
7. Check your solutions in the original equation.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-8
8
Completing the Square
Example Solve the equation x2 + 6x + 5 = 0 by
completing the square.
Step 1 Since the leading coefficient is 1, step 1 is not
necessary.
Step 2 Subtract 5 from both sides of the equation.
x  6x  5  0
2
x  6 x  5
2
continued
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Chapter 8-9
9
Completing the Square
Step 3 Determine the square of one-half the numerical
coefficient of the first degree term, 6.
1
2
(6)  3, 3  9
2
Add this value to both sides of the equation.
x  6 x  9  5  9
2
x  6x  9  4
2
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-10
10
Completing the Square
Step 4 By following this procedure, we produce a perfect
square trinomial on the left side of the equation. The
expression x2 + 6x + 9 is a perfect square trinomial that
can be factored as (x + 3)2.
( x  3)  4
2
Step 5 Use the square root property.
x3  4
x  3  2
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
continued
Chapter 8-11
11
Completing the Square
Step 6 Finally, solve for x by subtracting 3 from both
sides of the equation.
x  3  3  3  2
x  3  2
x  3  2 or
x  1
x  3  2
x  -5
Step 7 Check both solutions in the original equation.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 8-12
12