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 arrow’s speed, s, in feet per second, after t seconds is | -32t + 128 |. Find the values of t for which s is less that 48 feet per second. s = | -32t + 128 | < 48 Rewrite into 2 derived inequalities -32t + 128 < 48 -32t + 128 > -48 or x > 2.5 x < 5.5 Solve each inequality Check your answers -32(3) + 128 < 48 -32(5) + 128 > -48 -32 > -48 True! 32 < 48 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1

Quadratic Equation Problem 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? 13’ x s Let x = added length to each side of room (13 + x)2 = 250 A = s2  2.8

Aim: How do we solve quadratic equations by completing the square? Do Now: Evaluate a0 + a1/3 + a -2 when a = 8

Evaluate a0 + a1/3 + a -2 when a = 8 Evaluating Evaluate a0 + a1/3 + a -2 when a = 8 80 + 81/3 + 8-2 replace a with 8 1 + 81/3 + 8-2 x0 = 1 x1/3 = 1 + 2 + 8-2 x–n = 1/xn 8–2 = 1/82 = 1/64 1 + 2 + 1/64 3 1/64 combine like terms If m = 8, find the value of (8m0)2/3 (8 • 80)2/3 replace m with 8 (8 • 1)2/3 x0 = 1 (8)2/3 = 4

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 It has no negative exponents. It has no fractional exponents in the denominator. It is not a complex fraction. The index of any remaining radical is as small as possible.

Simplifying – Fractional Exponents

In a perfect square, there is a relationship Completing the Square Square of Binomial Perfect Square Trinomial (x + 3)2 = x2 + 6x + 9 (x - 4)2 = x2 - 8x + 16 (x - 7)2 = x2 - 14x + 49 (x - c)2 = x2 - 2cx + c2 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 x2 + 18x + c a perfect square. c = 81 x2 + 18x + 81 = (x + 9)2

To make the expression x2 + bx a perfect square, you must add Completing the Square Square of Binomial Perfect Square Trinomial = (half of b)2 The constant (3rd) term of the trinomial is the square of the coefficient of half the trinomial’s x-term. To make the expression x2 + bx a perfect square, you must add (1/2 b)2 to the expression.

Solving Quadratics by Completing the Square Complete the square and solve x2 - 6x = 40 Take 1/2 the coefficient of the linear term & square it. = (-3)2 = 9 Add the c term to both sides of equation x2 - 6x = 40 + 9 + 9 Binomial Squared (x - 3)2 = 49 Find square root of both sides Solve for x x - 3 = ±7 x - 3 = 7 x - 3 = -7 x = 10 x = -4 Graph this equation

Solving Quadratics when a  1 Complete the square and solve 4x2 + 2x - 5 = 0 Rewrite the original equation by adding 5 4x2 + 2x = 5 Divide by the equation by a (4) 4x2 + 2x = 5 4 = x2 + 1/2x = 5/4 Add 1/16 to each side x2 + 1/2x = 5/4 + 1/16 + 1/16 Binomial squared (x + 1/4)2 = 21/16 Find square root of both sides Solve for x Graph this equation

Aim: How do we solve quadratic equations by completing the square? Do Now: Find the value of c that makes x2 + 16x + c a perfect square. Square of Binomial Perfect Square Trinomial = (half of b)2

Completing the Square Problem 1 - 2 Find the value of c that makes x2 + 16x + c a perfect square. (half of 16)2 = (8)2 = 64 2. Solve by completing the square. x2 + 6x = 16 a. x2 - 4x + 2 = 0 b. x2 + 6x + 9 = 16 + 9 x2 - 4x = -2 x2 - 4x + 4 = -2 + 4 (x + 3)2 = 25 (x - 2)2 = 2 x + 3 = ±5 x + 3 = 5 x + 3 = -5 x = 2 x = -8 Graph these equation

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 x2 + (x + 12)2 = 602 x2 + x2 + 24x + 144 = 3600 2x2 + 24x + 144 = 3600 2 Pythagorean theorem x2 + 12x + 72 = 1800

Completing the Square Problem 3 (con’t) x2 + 12x + 72 = 1800 x2 + 12x = 1800 - 72 60” x2 + 12x = 1728 x2 + 12x + 36 = 1728 + 36 SONY (x + 6)2 = 1764 x + 6 =  42 x + 6 = 42 x + 6 = -42 x = 36 x = -48 Width = 36” Length = 36” + 12” = 48” Graph this equation