aim: completing the square course: adv. alg. & trig aim: how do we solve quadratic equations by...
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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 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
x > 2.5 x < 5.5Solve each inequality
Check your answers -32(3) + 128 < 48 -32(5) + 128 > -48
-32 > -48True! 32 < 48
0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1
-32t + 128 < 48 -32t + 128 > -48or
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 lengthto each side of room
13’
13’ x
x
(13 + x)2 = 250 A = s2
(13 x)2 250
13 x 250
x 250 13
s
s
Quadratic Equation Problem
2.8
Aim: Completing the Square Course: Adv. Alg. & Trig
Aim: How do we solve quadratic equations by completing the square?
Evaluate a0 + a1/3 + a -2 when a = 8
Do Now:
Aim: Completing the Square Course: Adv. Alg. & Trig
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 =
x3
83 21 + 2 + 8-2
x–n = 1/xn 8–2 = 1/82 = 1/641 + 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)2/3
(8 • 1)2/3 x0 = 1
= 4
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 Expression1. 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.
Aim: Completing the Square Course: Adv. Alg. & Trig
Simplifying – Fractional Exponents
3 5
4 4m n
1 1 5
3 6 12a b c
3 42 2
Aim: Completing the Square Course: Adv. Alg. & Trig
Completing the Square
Square of Binomial Perfect Square Trinomial
(x + 3)2 = x2 + 6x + 9
(x - 4)2 = x2 - 8x + 16
(x - c)2 = x2 - 2cx + c2
(x - 7)2 = x2 - 14x + 49
In 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 makes x2 + 18x + c a perfect square.
x2 + 18x + 81 = (x + 9)2
c = 81
Aim: Completing the Square Course: Adv. Alg. & Trig
Completing the Square
Square of BinomialPerfect Square Trinomial
=
x2 bx (b
2)2
(half of b)2
(x b
2)2
The constant (3rd) term of the trinomialis 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.
Aim: Completing the Square Course: Adv. Alg. & Trig
Solve for x
Find square rootof both sides
Binomial Squared
Add the c term to both sides of equation
Take 1/2 the coefficientof the linear term & square it.
Solving Quadratics by Completing the Square
Complete the square and solve x2 - 6x = 40
x2 - 6x = 40
(6
2)2
+ 9 + 9
(x - 3)2 = 49
(x 3)2 49
x - 3 = ±7
x - 3 = 7 x - 3 = -7x = 10 x = -4
= (-3)2 = 9
Graph this equation
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 rootof both sides
Solve for x
Add 1/16 to each side
Solving Quadratics when a 1
Complete the square and solve 4x2 + 2x - 5 = 0
(x + 1/4)2 = 21/16
x2 + 1/2x = 5/4
(1
4)2
+ 1/16 + 1/16
4x2 + 2x = 5
4x2 + 2x = 54
= x2 + 1/2x = 5/4
1
16
x 1
4
21
16
21
4
x 1
4
21
4Graph this equation
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 x2 + 16x + c a perfect square.
Do Now:
Square of BinomialPerfect Square Trinomial
=
x2 bx (b
2)2
(half of b)2
(x b
2)2
Aim: Completing the Square Course: Adv. Alg. & Trig
(half of 16)2
x2 + 6x = 16a.
Completing the Square Problem 1 - 2
1. Find the value of c that makes x2 + 16x + c a perfect square.
= 64
2. Solve by completing the square.
x2 + 6x + 9 = 16 + 9
(x + 3)2 = 25
x + 3 = ±5
x + 3 = 5 x + 3 = -5x = 2 x = -8
x2 - 4x + 2 = 0b.
x2 - 4x = -2
(x - 2)2 = 2
= (8)2
x2 - 4x + 4 = -2 + 4
x 2 2
x 2 2 x 2 2
x 2 2 x 2 2Graph these equation
Aim: Completing the Square Course: Adv. Alg. & Trig
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.
SONY
60”
Let x = width of TV x + 12 = length
x2 + (x + 12)2 = 602
x2 + x2 + 24x + 144 = 3600
2x2 + 24x + 144 = 3600
2x2 + 12x + 72 = 1800
Pythagorean theorem
Aim: Completing the Square Course: Adv. Alg. & Trig
Completing the Square Problem 3 (con’t)
SONY
60”
x2 + 12x + 72 = 1800
x2 + 12x = 1800 - 72
x2 + 12x = 1728
x2 + 12x + 36 = 1728 + 36(x + 6)2 = 1764
(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