Warm-Up Exercises Solve the equation. 1. ( )2)2 x 5 – 49= ANSWER 2 12, – 2. ( )2)2 x = ANSWER 526– –526– +, x 2x 2 18x Factor the expression. 4. x 2x 2 22x121 + – ANSWER ( )2)2 x 11 – ANSWER ( )2)2 x 9 +

Warm-Up Exercises 27 plus some number is 6 2. What is that number? ANSWER 9 5.

Example 1 Solve a Quadratic Equation Solve + x 2x 2 + 6x6x9 = 36. SOLUTION Write original equation. + x 2x 2 + 6x6x9 = 36 3 x+ Take the square root of each side. = 6 + – 3 x Solve for x. = 6 + – – Factor perfect square trinomial on left side. 3 2 x ( ( + 36 =

Example 1 Solve a Quadratic Equation ANSWER The solutions are and. 36+ – = 3 36 –– = 9 –

Example 2 Complete the Square Find the value of c that makes a perfect square trinomial. Then write the expression as the square of a binomial. x 2x 2 + 6x6xc– SOLUTION To find the value of c, complete the square using b = – 6.6. STEP 1Find half the coefficient of x ( ( – = – 3 STEP 2Square the result of Step ( ( – 9 = STEP 3Replace c with the result of Step 2. x 2x 2 + 6x6x9–

The trinomial is a perfect square when c 9. Then Example 2 Complete the Square x 2x 2 + 6x6xc– ANSWER = x 2x 2 + 6x6x9–x ( – = 3 2. (

Checkpoint 1. Solve by finding square roots. Perfect Square Trinomials x 2x 2 + 2x2x1– = 9 ANSWER 4,4, – 2 2. Find the value of c that makes a perfect square trinomial. Then write the expression as the square of a binomial. x 2x xc– ANSWER 36;x ( – 6 2 (

Example 3 Solve a Quadratic Equation Solve by completing the square. SOLUTION Write original equation. 2x 22x 2 4x4x6 + 0 = – 2x 22x 2 4x4x6 + 0 = – Divide each side by the coefficient of x 2. x 2x 2 2x2x3 + 0 = – Write the left side in the form x 2 bx. x 2x 2 2x2x3 = – + – Write the left side as the square of a binomial. 2 = – ()2)2 1x – ()2)2 1 – = – 2 = x 2x 2 2x2x1 + – 3 = – 1 + Add to each side.

Example 3 Solve a Quadratic Equation Take the square root of each side. = 1x – 2 – – + Add 1 to each side. = x 2 – – + 1 Write in terms of i. = x + – 1 i 2 ANSWER The solutions are and. +1 i 2 1 i 2 –

Checkpoint Solve the equation by completing the square. Solve a Quadratic Equation by Completing the Square x 2x 2 4x4x2 + 0 = – x 2x 2 8x8x = 5. 3w23w2 6w6w12 – = w 2w 2 12w4 – = 0 + ANSWER + – 2 6 – + – 4 13 – ANSWER + – 1 i 3 + – 6 4 2

Example 4 Write a Quadratic Function in Vertex Form Write in vertex form. Then identify the vertex. = x 2x 2 10x+22 – y SOLUTION = x 2x 2 10x+22 – y Write original equation. = y?+22x 2x 2 10x+? – () + Prepare to complete the square. = y25+22x 2x 2 10x+25 – () + Add to each side. ()2)2 5 – = – 2 = = y ()2)2 5x – Write as x 2x 2 10x+25 – ()2)2 5x –.

Example 4 Write a Quadratic Function in Vertex Form = y3 ()2)2 5x –– Solve for y. ANSWER The vertex form is. The vertex is. = y3 ()2)2 5x ––– () 35,5, You can check your answer by graphing the original equation.

Example 5 Use a Quadratic Equation to Model Area Construction A contractor is building a deck onto the side of a house. The deck will be a rectangle with an area of 1 20 square feet. The contractor has 32 feet of railing to use along 3 sides of the deck. Each side will be at least 8 feet long. What should the length and width of the deck be? SOLUTION – () 322x2xx 120 = length width area = – 322x 22x = Use the distributive property.

Example 5 Use a Quadratic Equation to Model Area – 16xx 2x 2 60 = – 4 = ()2)2 8x – Write left side as the square of a binomial. 2 = 8x – + – Take the square root of each side. x 10 = or 6 Solve for x. Divide each side by 2. – Reject the solution x 6 because the sides of the deck are at least 8 feet long. = – 16xx 2x 2 60 = – Add to each side. = – 2

Example 5 Use a Quadratic Equation to Model Area ANSWER The width is 10 feet. The length is feet. = 322 – 12 () 10

Checkpoint 7. Use Completing the Square Write in vertex form. Then identify the vertex. = x 2x 2 8x8x+19 – y ANSWER = y3;3; ()2)2 4x – + () 4, Geometry A rectangle has a length of 3x and a width of x 2. The area of the rectangle is 72 square units. Find the length and width of the rectangle. ANSWER 12 units, 6 units