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 |
< 48
Rewrite into 2 derived inequalities
-32t < 48
-32t > -48 or
x > 2.5
x < 5.5
Solve each inequality
Check your answers
-32(3) < 48
-32(5) > -48
-32 > -48
True!
32 < 48 1 2 3 4 5 6 7 -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’ x s
Let x = added length
to each side of room
(13 + x)2 = 250
A = 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
Evaluating
Evaluate a0 + a1/3 + a -2 when a = 8 /
replace a with 8
1 + 81/
x0 = 1
x1/3 =
x–n = 1/xn
8–2 = 1/82 = 1/64
/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

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 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.

6. Simplifying – Fractional Exponents

7. 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 =
x x + 16
(x - 7)2 =
x x + 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 = (x + 9)2

8. 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.

9. 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

10. 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

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 Binomial
Perfect Square Trinomial =
(half of b)2

12. 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 =
x2 - 4x = -2
x2 - 4x + 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

13. 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 = 3600
2x2 + 24x = 3600 2
Pythagorean theorem
x2 + 12x + 72 = 1800

14. Completing the Square Problem 3 (con’t)
x2 + 12x + 72 = 1800
x2 + 12x =
60”
x2 + 12x = 1728
x2 + 12x + 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