Simplify – Do not use a calculator

1) √24

2) √80

4.6 Solving Quadratic Equations by

Completing the Square

Learning Target: I can solve equations by completing the

square

Perfect Square Trinomials

Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36

2( 5) 64x 5 8x

When you take the square root, You MUST consider the Positive and Negative answers.

5 8x 5 8x 5 5

13x 5 5

3x

PerfectSquare

On One side

Take Square Root

ofBOTH SIDES

2 ( 5) 64x

PerfectSquare

On One side

Take Square Root

ofBOTH SIDES

But what happens if you DON’T have a perfect square on one side…….

You make it a Perfect Square

Use the relations on next slide…

2( 6)x ( 2 ) To expand a perfect square binomial:

2 12 36x x 6x 26

We can use this relationship to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial.

2 _ _12 _x x 12 2 6 626 36

36

2x

Take ½ middle term

Then square it

The resulting trinomial is called a perfect square trinomial,

which can be factored into a perfect square binomial.

2 _ _18 _ _x x

18 2 92(9) 81

81 2( 9)x

1. 2 12 0x x

1. Make one side a perfect square

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 0x x ___ ___12 2 6

2(6) 36

36 362( 6)x 362( 6) 36x 6 6x

6 6x 6 6x 6 6

12x 6 6

0x

12

Perfect Square Trinomials

Create perfect square trinomials.

x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___

100

4

25/4

Steps to solve by completing the square

1.) If the quadratic does not factor, move theconstant to the other side of the equation Ex: x²-4x -7 =0 x²-4x=7

2.) Work with the x²+ x side of the equation and complete the square by taking ½ of the coefficientof x and squaring Ex. x² -4x 4/2= 2²=4

3.) Add the number you got to complete the square toboth sides of the equationEx: x² -4x +4 = 7 +4

4.)Simplify your trinomial square Ex: (x-2)² =11

5.)Take the square root of both sides of the equationEx: x-2 =±√11

6.) Solve for xEx: x=2±√11

Solving Quadratic Equations by Completing the Square

Solve the following equation by completing the square:

Step 1: Set quadratic equation equal to zero 2 8 20 0x x

2 8 20x x

Solving Quadratic Equations by Completing the Square

Step 2: Find the term that completes the square. Add that term that is equal to zero into the equation.

X2 + 8x + ____ + _____ -20 = 016 -16

Solving Quadratic Equations by Completing the Square

Step 3: Factor the terms that create the perfect square trinomial. Simplify the other 2 terms of the equation.

X2 + 8x + ____ + _____ -20 = 016 -16

(x + 4)(x + 4) - 36 = 0

(x + 4)2 - 36 = 0

Note: This is vertex form of the equationy =(x + 4)2 - 36

Solving Quadratic Equations by Completing the Square

Step 4: Move the constant term and isolate the square binomial.

(x + 4)2 - 36 = 0 (x + 4)2 = 36

Solving Quadratic Equations by Completing the Square

Step 5: Take the square root of each side

2( 4) 36x

( 4) 6x

Solving Quadratic Equations by Completing the Square

Step 6: Set up the two possibilities and solve

4 6

4 6 an

d 4 6

10 and 2 x=

x

x x

x

Was there an easier way?2 8 20x x

Solve by Completing the Square2 6 16 0x x

2 6 16x x +9 +9

2 6 9 25x x 2

3 25x 3 5x

3 5x 8x 2x

Solve by Completing the Square

2 22 21 0x x 2 22 21x x

+121 +1212 22 121 100x x

211 100x

11 10x 11 10x 21x 1x

Solve by Completing the Square

2 2 5 0x x 2 2 5x x

+1 +12 2 1 6x x

21 6x

1 6x 1 6x

Solve by Completing the Square

2 10 4 0x x 2 10 4x x

+25

+252 10 25 29x x

25 29x 5 29x

5 29x

Solve by Completing the Square

01182 xx1182 xx

+16

+16 51682 xx

54 2 x 54 x

54 x

Solve by Completing the Square

0462 xx462 xx

+9 +9

5962 xx 53 2 x 53 x

53x

Assignment pg 237-238

Homework– p. 237 1-11

Challenge - 76

Completing the Square-Example #2

Solve the following equation by completing the square:

Step 1: If the lead coefficient is not 1, factor the lead coefficient from the a and b terms.

2x2 + 12x - 5 = 0

2(x2 + 6x) - 5 = 0

Solving Quadratic Equations by Completing the Square

Step 2: Find the term that completes the square.

(remember add in zero)

The quadratic coefficient must be equal to 1 before you complete the square, so you must divide the first 2 terms by the quadratic coefficient first.

2(x2 + 6x + ___) +___ - 5 = 09

Solving Quadratic Equations by Completing the Square

Step 2: What makes out of the parenthesis zero?

2(x2 + 6x + ___) +___ - 5 = 09 -18

Solving Quadratic Equations by Completing the Square

Step 3: Factor the perfect square trinomial in the equation. Combine the other two terms.

2(x2 + 6x + ___) +___ - 5 = 0

2(x + 3)2 - 23 = 0

9 -18

Solving Quadratic Equations by Completing the Square

Step 4: Move the constant to the right side of the equation and solve.

Isolate the perfect square

Take the square root of both sides

 

 

Solving Quadratic Equations by Completing the Square

Step 4 continued: 

Solving Quadratic Equations by Completing the Square

Try the following examples. Do your work on your paper and then check your answers.

1. x2 + 2x - 63 = 0

2. x2 - 10x - 15 = 0

3. 2x2 - 6x - 1 = 0