1 2 Objective Solve quadratic equations by completing

1. 2.

Objective Solve quadratic equations by completing the square.

“b divided by two, squared” X 2 + 6 x + 9 x 2 – 8 x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.

An expression in the form x 2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x 2 + bx to form a trinomial that is a perfect square. This is called completing the square.

Example 1: Completing the Square Complete the square to form a perfect square trinomial. A. x 2 + 2 x + x 2 + 2 x B. x 2 – 6 x + Identify b. x 2 + – 6 x . x 2 + 2 x + 1 x 2 – 6 x + 9

Check It Out! Example 1 Complete the square to form a perfect square trinomial. c. x 2 + 12 x + x 2 + 12 x d. 8 x + x 2 + Identify b. x 2 + 8 x . x 2 + 12 x + 36 x 2 + 12 x + 16

Solving a Quadratic Equation by Completing the Square

Example 2 A: Solving x 2 +bx = c Solve by completing the square. x 2 + 16 x = – 15 Step 1 x 2 + 16 x = – 15 Step 2 The equation is in the form x 2 + bx = c. . Step 3 x 2 + 16 x + 64 = – 15 + 64 Complete the square. Step 4 (x + 8)2 = 49 Factor and simplify. Step 5 x + 8 = ± 7 Take the square root of both sides. Write and solve two equations. Step 6 x + 8 = 7 or x + 8 = – 7 x = – 1 or x = – 15

Example 2 A Continued Solve by completing the square. x 2 + 16 x = – 15 The solutions are – 1 and – 15. Check x 2 + 16 x = – 15 (– 1)2 + 16(– 1) 1 – 16 – 15 x 2 + 16 x = – 15 (– 15)2 + 16(– 15) 225 – 240 – 15

Example 2 B: Solving x 2 +bx = c Solve by completing the square. x 2 – 4 x – 6 = 0 Step 1 x 2 + (– 4 x) = 6 Step 2 Write in the form x 2 + bx = c. . Step 3 x 2 – 4 x + 4 = 6 + 4 Complete the square. Step 4 (x – 2)2 = 10 Factor and simplify. Take the square root of both sides. Step 6 x – 2 = √ 10 or x – 2 = –√ 10 Write and solve two x = 2 + √ 10 or x = 2 – √ 10 equations. Step 5 x – 2 = ± √ 10

Example 2 B Continued Solve by completing the square. The solutions are 2 + √ 10 and x = 2 – √ 10. Check Use a graphing calculator to check your answer.

Check It Out! Example 2 a Solve by completing the square. x 2 + 10 x = – 9 Step 1 x 2 + 10 x = – 9 Step 2 Step 3 x 2 + 10 x + 25 = – 9 + 25 Step 4 (x + 5)2 = 16 Step 5 x + 5 = ± 4 Step 6 x + 5 = 4 or x + 5 = – 4 x = – 1 or x = – 9 The equation is in the form x 2 + bx = c. . Complete the square. Factor and simplify. Take the square root of both sides. Write and solve two equations.

Check It Out! Example 2 a Continued Solve by completing the square. x 2 + 10 x = – 9 The solutions are – 9 and – 1. Check x 2 + 16 x = – 15 (– 1)2 + 16(– 1) 1 – 16 – 15 x 2 + 10 x = – 9 (– 9)2 + 10(– 9) – 9 81 – 90 – 9 – 9

Example 3 A: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. – 3 x 2 + 12 x – 15 = 0 Divide by – 3 to make a = 1. Step 1 x 2 – 4 x + 5 = 0 x 2 – 4 x = – 5 x 2 + (– 4 x) = – 5 Step 2 Write in the form x 2 + bx = c. . Step 3 x 2 – 4 x + 4 = – 5 + 4 Complete the square.

Example 3 A Continued Solve by completing the square. – 3 x 2 + 12 x – 15 = 0 Step 4 (x – 2)2 = – 1 Factor and simplify. There is no real number whose square is negative, so there are no real solutions.

Example 4: Problem-Solving Application A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary. 1 Understand the Problem The answer will be the length and width of the room. List the important information: • The room area is 195 square feet. • The width is 2 feet less than the length.

Example 4 Continued 2 Make a Plan Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation.

Example 4 Continued 3 Solve Let x be the width. Then x + 2 is the length. Use the formula for area of a rectangle. l • w = A length x+2 times • width x = area of room = 195

Example 4 Continued Step 1 x 2 + 2 x = 195 Step 2 Simplify. . Step 3 x 2 + 2 x + 1 = 195 + 1 Complete the square by adding 1 to both sides. Step 4 (x + 1)2 = 196 Factor the perfect-square trinomial. Take the square root of Step 5 x + 1 = ± 14 both sides. Step 6 x + 1 = 14 or x + 1 = – 14 Write and solve two equations. x = 13 or x = – 15

Example 4 Continued Negative numbers are not reasonable for length, so x = 13 is the only solution that makes sense. The width is 13 feet, and the length is 13 + 2, or 15, feet. 4 Look Back The length of the room is 2 feet greater than the width. Also 13(15) = 195.

Lesson Quiz: Part I Complete the square to form a perfect square trinomial. 1. x 2 +11 x + 2. x 2 – 18 x + 81 Solve by completing the square. 3. x 2 – 2 x – 1 = 0 4. 3 x 2 + 6 x = 144 5. 4 x 2 + 44 x = 23 6, – 8

Lesson Quiz: Part II 6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft 2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner? 8 feet by 15 feet