Simplify – Do not use a calculator 1) √24 2) √80 Hint: Start by finding the Prime Factorization of each number

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

Perfect Square On One side Review Solve for x Take Square Root of BOTH SIDES When you take the square root, You MUST consider the Positive and Negative answers.

But what happens if you DON’T have a perfect square on one side……. Solve for x Take Square Root of BOTH 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…

To expand a perfect square binomial: Short Cut To expand a perfect square binomial: 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. Then

Make this a perfect square trinomial 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.

Solve by completing the square Make one side a perfect square Add a blank to both sides Divide “b” by 2 4. Square that answer. Add it to both sides Factor 1st side Square root both sides Solve for x Solve by completing the square 1.

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 the constant 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 coefficient of x and squaring Ex. x² -4x 4/2= 2²=4 3.) Add the number you got to complete the square to both sides of the equation Ex: 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 equation Ex: x-2 =±√11 6.) Solve for x Ex: 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

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 = 0 16 -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 = 0 16 -16 (x + 4)(x + 4) - 36 = 0 (x + 4)2 - 36 = 0 Note: This is vertex form of the equation y =(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

Solving Quadratic Equations by Completing the Square Step 6: Set up the two possibilities and solve Was there an easier way?

Solve by Completing the Square +9

Solve by Completing the Square +121

Solve by Completing the Square +1

Solve by Completing the Square +25

Solve by Completing the Square +16

Solve by Completing the Square +9

Assignment pg 237-238 Homework– p. 237 1-17 odds

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 = 0 9

Solving Quadratic Equations by Completing the Square Step 2: What makes out of the parenthesis zero? 2(x2 + 6x + ___) +___ - 5 = 0 9 -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. x2 + 2x - 63 = 0 x2 - 10x - 15 = 0 2x2 - 6x - 1 = 0