Presentation on theme: "Aim: How do we solve quadratic equations by completing the square?"— Presentation transcript:
1 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 |. Find the values of t for which s is less that 48 feet per second.s = | -32t |< 48Rewrite into 2 derived inequalities-32t < 48-32t > -48orx > 2.5x < 5.5Solve each inequalityCheck your answers-32(3) < 48-32(5) > -48-32 > -48True!32 < 481234567-7-6-5-4-3-2-1
2 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’xsLet x = added lengthto each side of room(13 + x)2 = 250A = s2 2.8
3 Aim: How do we solve quadratic equations by completing the square? Do Now:Evaluate a0 + a1/3 + a -2 when a = 8
4 Evaluate a0 + a1/3 + a -2 when a = 8 EvaluatingEvaluate a0 + a1/3 + a -2 when a = 8/replace a with 81 + 81/x0 = 1x1/3 =x–n = 1/xn8–2 = 1/82 = 1/64/643 1/64combine like termsIf m = 8, find the value of (8m0)2/3(8 • 80)2/3replace m with 8(8 • 1)2/3x0 = 1(8)2/3= 4
5 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 ExpressionIt 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.
7 In a perfect square, there is a relationship Completing the SquareSquare of BinomialPerfect Square Trinomial(x + 3)2=x2 + 6x + 9(x - 4)2=x x + 16(x - 7)2=x x + 49(x - c)2=x2 - 2cx + c2In a perfect square, there is a relationshipbetween the coefficient of the middle termand the constant (3rd) term.Describe it.Find the value of the c that makesx2 + 18x + c a perfect square.c = 81x2 + 18x = (x + 9)2
8 To make the expression x2 + bx a perfect square, you must add Completing the SquareSquare of BinomialPerfect Square Trinomial=(half of b)2The constant (3rd) term of the trinomialis the square of the coefficient of half thetrinomial’s x-term.To make the expression x2 + bxa perfect square, you must add(1/2 b)2 to the expression.
9 Solving Quadratics by Completing the Square Complete the square and solvex2 - 6x = 40Take 1/2 the coefficientof the linear term &square it.= (-3)2 = 9Add the c term toboth sides of equationx2 - 6x = 40+ 9+ 9Binomial Squared(x - 3)2 = 49Find square rootof both sidesSolve for xx - 3 = ±7x - 3 = 7x - 3 = -7x = 10x = -4Graph this equation
10 Solving Quadratics when a 1 Complete the square and solve 4x2 + 2x - 5 = 0Rewrite the original equation by adding 54x2 + 2x = 5Divide by theequation by a (4)4x2 + 2x = 54= x2 + 1/2x = 5/4Add 1/16 to each sidex2 + 1/2x = 5/4+ 1/16+ 1/16Binomial squared(x + 1/4)2 = 21/16Find square rootof both sidesSolve for xGraph this equation
11 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 BinomialPerfect Square Trinomial=(half of b)2
12 Completing the Square Problem 1 - 2 Find the value of c that makesx2 + 16x + c a perfect square.(half of 16)2= (8)2= 642. Solve by completing the square.x2 + 6x = 16a.x2 - 4x + 2 = 0b.x2 + 6x + 9 =x2 - 4x = -2x2 - 4x + 4 =(x + 3)2 = 25(x - 2)2 = 2x + 3 = ±5x + 3 = 5x + 3 = -5x = 2x = -8Graph these equation
13 Completing the Square Problem 3 Television screens are usually measured by thelength of the diagonal. An oversized televisionhas a 60-inch diagonal. The screen is 12 incheswider than its height. Find the dimensionsof the screen.SONY60”Let x = width of TVx + 12 = lengthx2 + (x + 12)2 = 602x2 + x2 + 24x = 36002x2 + 24x = 36002Pythagorean theoremx2 + 12x + 72 = 1800