Warm-Up

Factor the following trinomials. What do you notice?

1. x2 - 6x + 9 2. 4x2 - 4x + 1

COMPLETING THE SQUARE

Today’s Objective

Today, you’ll learn how to solve equations by completing the square.

Perfect Square Trinomials

1) x2 – 8x + ____ 3) x2 + 16x + ____

2) x2 + 7x + ____ 4) x2 – 13x + ____

Can you write a formula that will work for every example?

1) x2 – 8x + ____ 3) x2 + 16x + ____

2) x2 + 7x + ____ 4) x2 – 13x + ____

____2 bxx

Introducing… the magic number!

2

2

b

2

2

b This will always

result in a perfect square

trinomial!

to both sides of a quadratic equation is called completing the square.

Adding the magic number,

2

2

b

Example 1: Solve by Completing the Square

Solve x2 – 12x + 5 = 0.

First: x2 – 12x + ____ = -5

Example 2: Solve by Completing the Square

Solve x2 – 8x + 36 = 0.

Example 3: Solve by Completing the Square

Solve 5x2 = 6x + 8.

VERTEX FORM

Standard form vertex

Vertex: highest or lowest point on the graph.2 ways to find Vertex:1) Calculator: 2nd CALCMIN or MAX2) Algebraically

Find the Vertex1) x2 + 8x + 1

2) x2 + 2x – 5

3) 2x2 – 10x + 3

Complete the Square InvestigationStep 1: Complete the square:

X2 + 4x – 4 = 0

Step 2: DON’T SOLVE! Instead get zero on one side.

Step 3: graph the non-zero side and find the vertex

Completing the Square Finds the vertex!

Use completing the square to find the vertex of each:1) x2 + 6x + 8 = 0

2) X2 – 2x + 10 = 0

Vertex Form

Lucky for us, we can use a calculator to find the vertex instead of completing the square!

Converting from Standard to VertexStandard: y = ax2 + bx + cThings you will need:

a = and Vertex:

Vertex: y = a(x – h)2 + k

Example

Convert from standard form to vertex form.

y = -3x2 + 12x + 5

Example

Convert from standard form to vertex form.

y = x2 + 2x + 5

Now Convert and SolveConvert each quadratic from standard to vertex form. Then Solve for x.1. x2 + 6x – 5 = 0

Now Convert and Solve

Convert each quadratic from standard to vertex form.1. 3x2 – 12x + 7 = 0

2. -2x2 + 4x – 3 = 0