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Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by completing the square? An archer shoots an arrow into the air with an initial velocity of 128 feet per second. Because speed is the absolute value of velocity, the arrows speed, s, in feet per second, after t seconds is | -32t |. Find the values of t for which s is less that 48 feet per second. s = | -32t |< 48 Rewrite into 2 derived inequalities x > 2.5 x < 5.5 Solve each inequality Check your answers -32(3) < 48-32(5) > > -48 True!32 < t < 48-32t > -48 or

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Aim: Completing the Square Course: Adv. Alg. & Trig John is changing the floor plan of his home to include the dining room. The current dimensions of the room are 13 by 13. John wants to keep the square shape of the room and increase to total floor space to 250 square feet. How much will this add to the dimensions of the current room? Let x = added length to each side of room 13 x x (13 + x) 2 = 250 A = s 2 s s Quadratic Equation Problem 2.8

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Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by completing the square? Evaluate a 0 + a 1/3 + a -2 when a = 8 Do Now:

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Aim: Completing the Square Course: Adv. Alg. & Trig Evaluating Evaluate a 0 + a 1/3 + a -2 when a = / replace a with / x 0 = 1 x 1/3 = x –n = 1/x n 8 –2 = 1/8 2 = 1/ /64 3 1/64 combine like terms If m = 8, find the value of (8m 0 ) 2/3 (8 8 0 ) 2/3 replace m with 8 (8) 2/3 (8 1) 2/3 x 0 = 1 = 4

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Aim: Completing the Square Course: Adv. Alg. & Trig Simplifying – Fractional Exponents A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions. Conditions for a Simplified Expression 1.It has no negative exponents. 2.It has no fractional exponents in the denominator. 3.It is not a complex fraction. 4.The index of any remaining radical is as small as possible.

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Aim: Completing the Square Course: Adv. Alg. & Trig Simplifying – Fractional Exponents

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Aim: Completing the Square Course: Adv. Alg. & Trig Completing the Square Square of BinomialPerfect Square Trinomial (x + 3) 2 =x 2 + 6x + 9 (x - 4) 2 =x 2 - 8x + 16 (x - c) 2 =x 2 - 2cx + c 2 (x - 7) 2 =x x + 49 In a perfect square, there is a relationship between the coefficient of the middle term and the constant (3rd) term. Describe it. Find the value of the c that makes x x + c a perfect square. x x + 81 = (x + 9) 2 c = 81

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Aim: Completing the Square Course: Adv. Alg. & Trig Completing the Square Square of BinomialPerfect Square Trinomial = (half of b) 2 The constant (3rd) term of the trinomial is the square of the coefficient of half the trinomials x-term. To make the expression x 2 + bx a perfect square, you must add (1/2 b) 2 to the expression.

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Aim: Completing the Square Course: Adv. Alg. & Trig Solve for x Find square root of both sides Binomial Squared Add the c term to both sides of equation Take 1/2 the coefficient of the linear term & square it. Solving Quadratics by Completing the Square Complete the square and solve x 2 - 6x = (x - 3) 2 = 49 x - 3 = ±7 x - 3 = 7x - 3 = -7 x = 10x = -4 = (-3) 2 = 9 Graph this equation

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Aim: Completing the Square Course: Adv. Alg. & Trig Rewrite the original equation by adding 5 Divide by the equation by a (4) Binomial squared Find square root of both sides Solve for x Add 1/16 to each side Solving Quadratics when a 1 Complete the square and solve 4x 2 + 2x - 5 = 0 (x + 1/4) 2 = 21/16 x 2 + 1/2x = 5/4 + 1/16 4x 2 + 2x = 5 4 = x 2 + 1/2x = 5/4 Graph this equation

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Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by completing the square? Find the value of c that makes x x + c a perfect square. Do Now: Square of BinomialPerfect Square Trinomial = (half of b) 2

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Aim: Completing the Square Course: Adv. Alg. & Trig (half of 16) 2 x 2 + 6x = 16 a. Completing the Square Problem Find the value of c that makes x x + c a perfect square. = Solve by completing the square. x 2 + 6x + 9 = (x + 3) 2 = 25 x + 3 = ±5 x + 3 = 5x + 3 = -5 x = 2x = -8 x 2 - 4x + 2 = 0 b. x 2 - 4x = -2 (x - 2) 2 = 2 = (8) 2 x 2 - 4x + 4 = Graph these equation

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Aim: Completing the Square Course: Adv. Alg. & Trig Completing the Square Problem 3 Television screens are usually measured by the length of the diagonal. An oversized television has a 60-inch diagonal. The screen is 12 inches wider than its height. Find the dimensions of the screen. SONY 60 Let x = width of TV x + 12 = length x 2 + (x + 12) 2 = 60 2 x 2 + x x = x x = x x + 72 = 1800 Pythagorean theorem

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Aim: Completing the Square Course: Adv. Alg. & Trig Completing the Square Problem 3 (cont) SONY 60 x x + 72 = 1800 x x = x x = 1728 x x + 36 = (x + 6) 2 = 1764 x + 6 = 42 x + 6 = -42 x = 36 x = -48 Width = 36 Length = = 48 Graph this equation

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