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 + ____ x2 + 7x + ____ 4) x2 – 13x + ____

Can you write a formula that will work for every example? 1) x2 – 8x + ____ 3) x2 + 16x + ____ x2 + 7x + ____ 4) x2 – 13x + ____

This will always result in a perfect square trinomial! Introducing… the magic number! This will always result in a perfect square trinomial!

Adding the magic number, to both sides of a quadratic equation is called completing the square.

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  CALC MIN or MAX 2) Algebraically

Find the Vertex x2 + 8x + 1 x2 + 2x – 5 2x2 – 10x + 3

Complete the Square Investigation Step 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: x2 + 6x + 8 = 0 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 Vertex Standard: y = ax2 + bx + c Things 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 Solve Convert each quadratic from standard to vertex form. Then Solve for x. x2 + 6x – 5 = 0

Now Convert and Solve Convert each quadratic from standard to vertex form. 3x2 – 12x + 7 = 0 -2x2 + 4x – 3 = 0