(x + 3) 2 = (x + 3)(x + 3) = x 2 + 3x + 3x + 9 = x 2 + 6x + 9 Perfect Square Trinomials

Perfect Square Trinomials Perfect Square Trinomials are trinomials that have: 1. a product (ax 2 + bx + c) that is a perfect square 2. a sum that has the same factors of the product when added together (think: product sum). (x + 3) 2 = (x + 3)(x + 3) = x 2 + 3x + 3x + 9 = x 2 + 6x + 9 Perfect Square Trinomials Be able to recognize perfect square trinomials! Understand the pattern that they create

Perfect Square Trinomials (Positive Sum) Perfect Square Trinomials (Negative Sum)

Why Create a Perfect Square Trinomial? To be able to write the quadratic in a simplified form (vertex form) so we can solve the quadratic more easily. If we have x 2 + 4x + 4, then we can write (x + 2) 2 because we know that it is equivalent. Creating a Perfect Square Trinomial *The quadratic in the problem will not be a perfect square trinomial...we are going to create one to be able to solve it. Quadratic must be in standard form We can create a perfect square trinomial with an odd bx value (or with a #x 2 ), but it is much easier with a x 2 and an even bx value.

Steps to Creating a Perfect Square Trinomial 1. Move the product to the other side of the equals sign and leave a "+ " where the product was that was moved to the other side and put a "+ " next to the new location of the product. x 2 2x 24 = 0 2. Divide the 'b' value by 2. This is the value that will go into the parenthesis of the simplified form of the Perfect Square Trinomial. x 2 2x 24 = 0

3. Square the number that was found in the last step. This number goes in BOTH blanks. x 2 2x 24 = 0 We have justified writing the (x 1) 2 by putting the 1 in both blanks because the left side of the quadratic has become a perfect square trinomial, which can be also written as (x 1) 2. 4. Combine/add values on the right side of the equals sign. If it is not a perfect square, then the solution will have a radical in it. x 2 2x 24 = 0

Write the quadratic using perfect square trinomials. x 2 + 4x 8 = 0 Write the quadratic using perfect square trinomials. x 2 8x 10 = 0

This is why we only want to use perfect square trinomials when we have an even bx! Write the quadratic using perfect square trinomials. x 2 + 3x 6 = 0 This is why we only want to use perfect square trinomials if we have an x 2! Write the quadratic using perfect square trinomials. 3x 2 + 4x 18 = 0

Write the quadratic using perfect square trinomials. x 2 + 18x 15 = 0 Write the quadratic using perfect square trinomials. x 2 14x 15 = 0

Fill in the blanks of the values that make it a perfect square trinomials and its simplified form. 1. x 2 + 4x + = (x ) 2 2. x 2 8x + = (x ) 2 x 2 + 4x + 4 = (x + 2) 2 x 2 8x + 16 = (x 4) 2 3. x 2 6x + = (x ) 2 4. x 2 + 12x + = (x ) 2 x 2 6x + 9 = (x 3) 2 x 2 + 12x + 36 = (x + 6) 2 Write the quadratic using perfect square trinomials. 5. x 2 + 6x 16 = 0 6. x 2 10x + 8 = 0 (x + 3) 2 = 25 (x 5) 2 = 13 7. x 2 + 12x 20 = 0 8. x 2 2x 24 = 0 (x + 6) 2 = 55 (x 1) 2 = 25 Fill in the blanks of the values that make it a perfect square trinomials and its simplified form. 1. x 2 + 4x + = (x ) 2 2. x 2 8x + = (x ) 2 x 2 + 4x + 4 = (x + 2) 2 x 2 8x + 16 = (x 4) 2 3. x 2 6x + = (x ) 2 4. x 2 + 12x + = (x ) 2 x 2 6x + 9 = (x 3) 2 x 2 + 12x + 36 = (x + 6) 2 Write the quadratic using perfect square trinomials. 5. x 2 + 6x 16 = 0 6. x 2 10x + 8 = 0 (x + 3) 2 = 25 (x 5) 2 = 17 7. x 2 + 12x 20 = 0 8. x 2 2x 24 = 0 (x + 6) 2 = 56 (x 1) 2 = 25

Bell Ringer Write the quadratic using perfect square trinomials. 1. x 2 + 6x 55 = 0 2. x 2 + 18x + 19 = 0 + 55 +55 19 19 x 2 + 6x + 9 = 55 + 9 x 2 + 18x + 81 = 19 + 81 6/2 = 3 (x + 3) 2 = 64 18/2 = 9 (x + 9) 2 = 62 (3) 2 = 9 (9) 2 = 81 Why Create a Perfect Square Trinomial? To be able to write the quadratic in a simplified form (vertex form) so we can solve the quadratic more easily by using square roots.

Steps to Solving a Perfect Square Trinomial 1. Move the c (ax 2 + bx + c) to the other side of the equals sign and leave a "+ " where the c was that was moved to the other side and put a "+ " next to the new location of the c. 2. Divide the 'b' value by 2. This is the value that will go into the parenthesis of the simplified form of the Perfect Square Trinomial. 3. Square the number that was found in the last step. This number goes in BOTH blanks. 4. Combine/add values on the right side of the equals sign. If it is not a perfect square, then the solution will have a radical in it.

***NEW STEP*** 5. Solve the quadratic using square roots. x 2 2x 24 = 0 +24 +24 x 2 2x + 1 = 24 + 1 2/2 = 1 (x 1) 2 = 25 ( 1) 2 = 1 Solve the quadratic by completing the square. x 2 + 16x 17 = 0

Solve the quadratic by completing the square. x 2 6x + 33 = 40 Solve the quadratic by completing the square. x 2 + 8x 20 = 0

Solve the quadratics by completing the square. x 2 + 6x 55 = 0 x 2 + 18x + 32 = 0 Sometimes when solving the quadratic, the number on the right side of the equals sign is not a perfect square. Simplify the radical, if possible.

Solve the quadratics by completing the square. Put your answer in simplified radical form. x 2 10x 28 = 0 Solve the quadratic by completing the square. Put your answer in simplified radical form. x 2 4x 14 = 0

Solve the quadratics by completing the square. Put your answer in simplified radical form. 1. x 2 10x 40 = 0 2. x 2 + 14x + 20 = 3 Solve the quadratics by creating perfect square trinomials. Put your answers in simplified radical form. 1. x 2 + 2x 3 = 0 2. x 2 + 10x + 14 = 7 x = 1, 3 x = 3, 7 3. x 2 12x + 26 = 0 4. x 2 8x 48 = 0 x = 6 + 10 x = 12, 4 5. x 2 4x 20 = 0 6. x 2 + 16x 22 = 0 x = 2 + 2 6 x = 8 + 86 7. x 2 + 6x 9 = 7 8. x 2 + 20x + 25 = 0 x = 2, 8 x = 10 + 5 3

Solve the quadratics by creating perfect square trinomials. Put your answers in simplified radical form. 1. x 2 + 2x 3 = 0 2. x 2 + 10x + 14 = 7 x = 1, 3 x = 3, 7 3. x 2 12x + 26 = 0 4. x 2 8x 48 = 0 x = 6 + 10 x = 12, 4 5. x 2 4x 20 = 0 6. x 2 + 16x 22 = 0 x = 2 + 2 6 x = 8 + 86 7. x 2 + 6x 9 = 7 8. x 2 + 20x + 25 = 0 x = 2, 8 x = 10 + 5 3