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Completing the Square Section 9.1 MATH 116-460 Mr. Keltner

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Example 1 Solve ( x + 6) 2 = 20. We know how to solve equations when it is possible to take the square root of each side, such as: 4x 2 = 16 Constant multiple with a variable x 2 + 14x + 49 = 25 A quantity that can be rewritten x 2 = -76 Where we take the square root of a negative number, or use i

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A need for a new strategy Sometimes, we will have an expression of the form x 2 + bx, that requires us to insert an additional term to make it a perfect square, like (x + k) 2. The process of forcing a quadratic expression to become a perfect square trinomial is called completing the square.

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Completing the Square in 3 “Easy” Steps Steps These steps may only be used on a quadratic expression in the form ax 2 + bx, where a = 1, and b is a real number. 1. Find half the coefficient of x. 2. Square that value. 3. Replace c with the resulting value. Example: Find the value of c that makes x 2 -26x + c a perfect square trinomial. Then write the expression as the square of a trinomial. 1. -26 ÷ 2 = -13 2. (-13) 2 = 169 3. x 2 -26x + c becomes x 2 - 26x + 169, which is equal to ( x-13 ) 2 when it is factored.

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Example 2 Find the value of c that makes each expression a perfect square trinomial. Then write each expression as the square of a binomial. x 2 + 14x + c x 2 + 22x + c x 2 – 9x + c What do you notice about the value of c if b happens to be an odd number? Does the sign of c change or does it remain constant?

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Solving Quadratic Equations by Completing the Square Related example: Solve x 2 - 4 = 12. Your first step would be to add 4 to each side of the equation. Just the same as we add the same value to both sides of this equation, we apply this same idea when completing the square. Example 3 Solve x 2 - 10x + 1 = 0 by completing the square.

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Solving ax 2 + bx + c = 0 if a ≠ 1 Divide every term of both sides by the coefficient of x 2 (the value of a ). Make sure to balance the equation by adding the same value to both sides. Example 4: Solve 3x 2 – 36x + 150 = 0 by completing the square.

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Example 5 Solve the equation by completing the square. ( NOTE : You cannot apply the Zero-Product Property and say that either 6x = 12 or (x + 8) = 12 and solve.) 6x (x + 8) = 12

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Baseball example: Finding a maximum value The height, y (in feet), of a baseball x seconds after it is hit is given by the equation: y = -16x 2 + 96x + 3 Find the maximum height of the baseball. The maximum height of the baseball will be the y-coordinate of the vertex of the parabola. It will help if we can write the equation in vertex form.

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Baseball Example, Cont. The height, y (in feet), of a baseball x seconds after it is hit is given by the equation: y = -16x 2 + 96x + 3 Find the maximum height of the baseball. Start by writing the function in vertex form. If we can find the y-value at the vertex, we will have found the maximum height of the ball.

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Fountain Example At the Buckingham Fountain in Chicago, the water’s height h (in feet) above the main nozzle can be modeled by h = -16t 2 + 89.6t, where t is the time in seconds) since the water has left the nozzle. Find the highest point the water reaches above the fountain. What does this vertex represent, in real-world terms?

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Assessment Pgs. 628-631: #’s 7-98, multiples of 7

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