5.8 Solving Quadratic Funtions by Completing the Square

1/28/2013

VocabularyPerfect Square

Trinomial:Whenever you multiply a binomial by itself, the resulting trinomial is called a perfect square trinomial

12)1)(1(1 22 xxxxxExample:

x+1

x+1

This is the area of this square

VocabularyCompleting the

Square :The process of adding a constant c to the expression x2 + bx to make it a perfect square trinomial (PST)

How? By adding to x2 + bx 2

2

b

What is it used for:

Converting equations from standard form to vertex form.To solve quadratic functions when “Big X” does not work!

Factored form of PST:

The square of 2 binomials2

2

bx

Review

Equation of a Parabola in VERTEX FORM:

Where (h, k) is the vertex

khxay 2)(

Steps for completing the square

y = x2 + bx + c

y = (x2 + bx + + c -

Standard form : y = x2 + bx + c( )

y = (x + 2 + d1. Put ( ) around x2 + bx and move c

outside ( )2. Take half of b and square it. Add it

to the ( ) and subtract it from c.3. Factor the PST in ( ) and simplify c -Note: if y = x2 – bx + c

Then y = (x - )2 + d

This is a PST

Factored form of PST:

khxay 2)(

Example 1

y = x2 + 6x + 5

y = (x2 + 6x + + 5 -

Rewrite y = x2 + 6x + 5 in Vertex Form and determine the vertex.

( )

y = (x + 2 - 4

1. Put ( ) around x2 + 6x and move +5 outside ( )

2. Take half of 6 and square it. Add 9 to the ( ) and subtract 9 from 5.

3. Factor the PST in ( ) and simplify

Vertex (-3, -4)

This is a PST

Factored form of PST:

khxay 2)(

Example 2

y = x2 - 6x + 10

y = (x2 - 6x + + 10 -

Rewrite y = x2 - 6x + 10 in Vertex Form and determine the vertex.

( )

y = (x - 2 + 1

1. Put ( ) around x2 - 6x and move +10 outside ( )

2. Take half of 6 and square it. Add 9 to the ( ) and subtract 9 from 10.

3. Rewrite what’s in the ( ) as (x - 3)2

Vertex (3, 1)

Checkpoint Use Completing the Square

Write in vertex form. Then identify the vertex.

= x 2 8x + 19–y

ANSWER =y 3;( )24x – + ( )4, 3

Homework:

5.8 p.271 #15-20