Secondary Math 2 4.3 Complete the Square

Warm Up Use the quadratic formula to solve. 3 𝑥 2 −13𝑥+4=0

Recap of this unit so far… We now have two different methods to solve problems in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0. We can… …factor and solve. …use the quadratic formula. There is one final method! (All three are important to know how to do.)

What you will learn The third method to solving quadratics: complete the square.

Introduction… Using any method, solve 𝑥−2 2 −25=0 𝑥−2 𝑥−2 −25=0 𝑥 2 −4𝑥+4−25=0 𝑥 2 −4𝑥−21=0 𝑥+3 𝑥−7 =0 𝑥=−3, 𝑥=7

Introduction… Using any method, solve 𝑥−2 2 −25=0 𝑥−2 𝑥−2 −25=0 𝑥 2 −4𝑥+4−25=0 𝑥 2 −4𝑥−21=0 𝑥= 4± −4 2 −4 1 −21 2(1) 𝑥=−3, 𝑥=7

The easy way! Solve 𝑥−2 2 −25=0 𝑥−2 2 =25 𝑥−2 2 = 25 𝑥−2=±5 𝑥=2±5 𝑥=−3,𝑥=7

Today we will learn how to write quadratics in this form. From the introduction, we saw that 𝑥−2 2 −25=0 is the same as 𝑥 2 −4𝑥−21=0

By the end of class today, you will be able to write a quadratic function of the form 𝑥 2 −4𝑥−21=0 into 𝑥−2 2 −25=0.

Standard Form: y=𝑎 𝑥 2 +𝑏𝑥+𝑐 Vertex Form: 𝑦=𝑎 𝑥−ℎ 2 +𝑘

Practice some multiplication! 𝑥+4 2 = 𝑟+5 2 = 𝑧−7 2 = 𝑛−1 2 = Look for patterns.

These quadratics are called perfect squares.

Hint! Cut b in half, then square it. Or find 𝑏 2 2 Complete the square. 𝑟 2 −12𝑟+_____ Hint! Cut b in half, then square it. Or find 𝑏 2 2 Now write it as a perfect square. Try problems 2-4.

4) 𝑦 2 +42𝑦+________

5) 𝑧 2 −7𝑧+_____ Try problems 6-8

6) 𝑛 2 + 23 12 𝑛+______

How to complete the square for quadratic equations (where a=1) Step 1: Move the constant c to the other side of the equation. (we will do #10 as our example.) 𝑝 2 +12𝑝+27=0 𝑝 2 +12𝑝=−27

How to complete the square for quadratic equations (where a=1) Step 2: Complete the square on the left side of the equation. Balance this by adding the same value to the right side of the equation. 𝑝 2 +12𝑝=−27 𝑝 2 +12𝑝+____=−27+____ 𝑝 2 +12𝑝+36=−27+36

How to complete the square for quadratic equations (where a=1) Step 3: Write the left side of the equation as a perfect square. Evaluate the right side of the equation. 𝑝 2 +12𝑝+36=−27+36 𝑝+6 2 =−27+36 𝑝+6 2 =9

How to complete the square for quadratic equations (where a=1) Step 4 (solving): Solve for x. You should get two solutions. Done! 𝑝+6 2 =9

How to complete the square for quadratic equations (where a=1) Step 4 (vertex form): Rewrite the equation so that it equals 0. Done! 𝑝+6 2 =9 𝑝+6 2 −9=0

9) 𝑎 2 +4𝑎−15=0

12) 𝑎 2 −14𝑎+13=0

13) 𝑥 2 −3𝑥−66=4

Exit Problem Write the equation in vertex form (by completing the square). 𝑥 2 −20𝑥+12=0