Completing the Square 1.7

1. Completing theSquare1.7 What is completing the square? What steps do you follow to complete the square?

2. ANSWER The solutions are 4 + 5 = 9 and 4 –5 = – 1. Solve a quadratic equation by finding square roots Solve x2 – 8x + 16 = 25. x2 – 8x + 16 = 25 Write original equation. (x –4)2= 25 Write left side as a binomial squared. x – 4 = +5 Take square roots of each side. x = 4 +5 Solve for x.

3. Solve: (a + b)2 = (a - b)2 =

4. 16 = 8 2 Make a perfect square trinomial Find the value of cthat makes x2 + 16x + ca perfect square trinomial. Then write the expression as the square of a binomial. SOLUTION STEP 1 Find half the coefficient of x. STEP 2 Square the result of Step 1. 82 = 64 STEP 3 Replace c with the result of Step 2. x2 + 16x + 64 Thenx2 + 16x + 64 = (x + 8)(x + 8) = (x + 8)2

5. 22 = 11 2 Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. x2 + 22x + c x 11 x SOLUTION 11 STEP 1 Find half the coefficient of x. STEP 2 Square the result of Step 1. 112 = 121 STEP 3 Replace c with the result of Step 2. x2 + 22x + 121 The trinomialx2 + 22x + c is a perfect square whenc = 121.

6. Find the value of c that makesx2 – 6x + ca perfect square trinomial. Write the expression as the square of a binomial.

7. Steps for Completing the Squareax2 + bx + c 1.Make sure the coefficient of the x2 term is one. (If it is not, divide the equation by the coefficient.) 2.Move the constant number to the other side. 3. Divide “b” by 2 4. Square the result from #2. 5. Add this number to both sides of the equation. 6. Factor and solve for x.

8. x – 6 = + 32 x = 6 + 32 x = 6 + 4 2 2 Simplify: 32 16 2 4 = = ANSWER 2 2 The solutions are 6 + 4 and6 – 4 to each side. Solve ax2 + bx + c = 0 when a = 1 Solve x2 – 12x + 4 = 0 by completing the square. Write original equation. x2 – 12x + 4 = 0 x2 – 12x = – 4 Write left side in the form x2 + bx. x2– 12x + 36 = – 4 + 36 (x – 6)2 = 32 Write left side as a binomial squared. Take square roots of each side. Solve for x.

9. Solving a Quadratic equation if the coefficient of x2 is 1 Solve by completing the square. x2 + 10x -3 = 0 x2 + 10x + ___ = 3 +___ x2 + 10x + 25 = 3 + 25 (x + 5)2 = 28

10. Solving a Quadratic Equation if the Coefficient of x2 is not 1 Solve by completing the square. 3x2 – 6x + 12 = 0 3x2− 6x + ___ = −12 + ___ 3/3x2− 6/3x + ___=−12/3+ ___ x2 −2x + ___=−4 + ___ (−2/2)2=1 x2 −2x + 1=−4 + 1 (x−1)2 = −3

11. ) ( –10 Add to each side. 2 25 (–5) 2 = = 2 ANSWER The vertex form of the function is y = (x – 5)2– 3. The vertex is (5, – 3). Write a quadratic function in vertex form Write y = x2 – 10x + 22 in vertex form. Then identify the vertex. y = x2 – 10x + 22 Write original function. y + ?= (x2–10x + ?) + 22 Prepare to complete the square. y + 25= (x2– 10x + 25) + 22 y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared. y = (x – 5)2– 3 Solve for y.

12. Writing a Quadratic Function in Vertex Form Write the quadratic function in vertex form. y = x2 -8x +11 y + ___ = (x2−8x + ___) +11 y + 16 = (x2−8x + 16) +11 y + 16 = (x−4)2 +11 y = (x−4)2 −5 What is the vertex?

13. Find the maximum value of a quadratic function Baseball The height y(in feet) of a baseball tseconds after it is hit is given by this function: y = –16t2 + 96t + 3 Find the maximum height of the baseball. SOLUTION The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

14. What is completing the square? Another method for solving a quadratic equations. • What steps do you follow to complete the square? 1.Make sure the coefficient of the x2 term is one. (If it is not, divide the equation by the coefficient.) 2.Move the constant number to the other side. 3. Divide “b” by 2 4. Square the result from #2. 5. Add this number to both sides of the equation. 6. Factor and solve for x.

15. Assignment 1.7 p. 54, 3-7 odd, 13-17 odd, 23-27 odd, 39-45 odd, 51