Algebra 1 Common Core Student Edition, Grade 8-9

I

Pr o b l em 1 J

What is the value of c such that x 2 — 16x + c is a perfect-square trinomial?
Can c be negative?
No. c is th e square o f a The value of b is -16. The term to add to x2 - 16x is (“p i , or 64. So c = 64.
real number, w h ic h is

never negative. 1. What is the value of c such that x 2 + 20x + c is a perfect-square trinomial?

Fi n d i n g c t o Co m p l et e t h e Sq u a r e

To solve an equation in the form x2 + bx + c = 0, first subtract the constant term c

from each side of the equation.

Pr o b l em 2 So l v i n g x 2 + b x + c = 0

What are the solutions of the equation x 2 — 14x + 1 6 = 0?

Think
Why do you write
x - 7 « ±5.74 as
tw o equations?
Recall th a t th e sym bol ±
means "plus or minus."
That means x - 7
equals 5.74 or -5 .7 4.

x2 — 14x + 16 = 0

x2- 14x= -16 Subtract 16 from each side.

x2- 14x+ 49 = -16 + 49 Add ( j , or 49, to each side.

(x - 7)2 = 33 Write x2 - 14x + 49 as a sq uare.

x — 7 = + V33 Find square roots o f each side.

X - 7 » ± 5.74 Use a calculator to approxim ate V 3 3.

x - 7 ~ 5.74 or x-7 « -5.74 Write as two equations.

x = 5.74 + 7 or x « -5.74 + 7 Add 7 to each side.

x « 12.74 or x * 1.26 Simplify.

v J} Got It? 2. a. What are the solutions of the equation x2 + 9x + 15 = 0?

b. Reasoning Could you use factoring to solve part (a)? Explain.

The equation y = (x - h)2 + k represents a parabola with vertex (h, k). You can use

the method of completing the square to find the vertex of quadratic functions of the

form y = x2 + bx + c.

Thi nk

Why do you need to

add the same number

to each side?

You need to add th e

same number to each

side to keep the equation

balanced.

Pr o b l em 3 Fi n d i n g t h e V e r t e x b y Co m p l et i n g t h e Sq u a r e

Find the vertex of y = x2 + 6x + 8 by completing the square.

y - x2 + 6x + 8

y - 8 = x2 + 6x

y —8 + 9 = x 2 + 6x+9

y + 1 = (x + 3)2

y = (x + 3)2 — 1

Subtract 8 from each side.

Add ( | ) 2, or 9. to each side.

Simplify the left side and write the right side

as a square.

S u b tra c t 1 fr o m ea ch s id e so t h a t e q u a tio n is in v e rte x fo rm.

The vertex of y = x + 6x + 8 is (-3 , —1).

c

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Lesso n 9- 5 Co m p l et i n g t h e Sq u ar e 577