8.1 Completing a Square

Given: x2 = ux = +√(u) or x = -√(u)

E.g.Given: x2 = 3x = √(3) or x = -√(3)

Completing the Square Solve: x2 + 6x + 4 = 0 x2 + 6x = -4 How to make the left side a perfect square? x2 + 6x + 9 = -4 + 9 (x + 3)2 = 5 x + 3 = √5 or x + 3 = -√5 x = -3 + √5 or x = -3 - √5 Check:

(-3 + √5)2 + 6(-3 +√5) + 4 = 0 ?9 - 6√5 + 5 - 18 + 6√5 + 4 = 0 ?(9 + 5 – 18 + 4) + (-6√5 + 6√5 ) = 0 yes

Completing the Square Solve: 9x2 – 6x – 4 = 0 9x2 – 6x = 4 (9x2 – 6x )/9 = 4/9 x2 – (6/9)x = 4/9 x2 – (2/3)x + (1/9) = 4/9 + (1/9) (x – 1/3)2 = 5/9 x – 1/3 = √(5/9) or x – 1/3 = -√(5/9) x = 1/3 + √(5)/√(9) or x = 1/3 - √(5)/√(9)

x = 1/3 + √(5)/3 or x = 1/3 - √(5)/3 x = (1 + √(5))/3 or x = (1 - √(5))/3

Your Turn

Solve by completing the square:x2 + 3x – 1 = 0

x2 + 3x = 1 x2 + 3x + 9/4 = 1 + 9/4 (x + 3/2)2 = 13/4 x + 3/2 = ± √(13/4) x = -3/2 ± √(13)/2 x = (-3 ± √(13)) / 2

Your Turn Solve by completing the square:

3x2 + 6x + 1 = 0 3x2 + 6x = -1 (3x2 + 6x) / 3 = -1/3

x2 + 2x = -1/3 x2 + 2x + 1 = -1/3 + 1

(x + 1)2 = 2/3 x + 1 = ±√(2/3) x = -1 ±√(2/3) x = -1 ±√(2/3) √(3)/√(3) x = -1 ±√((2)(3 ))/ 3 x = -1 ±√(6) / 3

x = (-3 ±√(6) ) / 3

Review Solve by completing the square.

(4x – 1)2 = 15 16x2 – 8x + 1 = 15 16x2 – 8x = 14 (16x2 – 8x)/16 = 14/16 x2 – (1/2)x = 7/8 x2 – (1/2)x + 1/16 = 7/8 +1/16 (x – ¼)2 = 15/16 x – ¼ = ±√(15/16) x = ¼ + √(15/16) = ¼ + √(15)/4 = (1 + √(15))/4 x = ¼ - √(15/16) = ¼ - √(15)/4 = (1 - √(15))/4

8.2 Quadratic Formula Given: ax2 + bx + c = 0, where a > 0. ax2 + bx = -c (ax2 + bx)a = -c/a x2 + (b/a)x = -c/a x2 + (b/a)x + (b/2a)2 = -c/a + (b/2a)2

(x + (b/(2a))2 = -c/a + b2/(4a2) (x + (b/(2a))2 = -(c/a)(4a)/(4a) + b2/(4a2) (x + (b/(2a))2 = (-4ac) + b2) / (4a2) x + b/(2a) = ±√ ((b2 – 4ac)/(4a2))

x = (-b /(2a) ±√ ((b2 – 4ac)/(2a)x = (-b ±√ (b2 – 4ac)) / (2a)

Quadratic Formula Given: ax2 + bx + c = 0, where a > 0.

-b ± √(b2 – 4ac)x = ----------------------- 2a

E.g., if 3x2 – 2x – 4 = 0 a = 3; b = -2; c = -4

2 ±√(4 + 48) 2 ± √(52) 1 ± √(13) x = ---------------- = ------------ = --------------- 6 6 3

Application The number of fatal vehicle crashes has been

found to be a function of a driver’s age. Younger and older driver’s tend to be involved in more fatal accidents, while those in the 30’s and 40’s tend to have the least such accidents.

The number of fatal crashes per 100 million miles, f(x), as a function of age, x, is given by f(x) = 0.013x2 – 1.19x + 28.24

What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven?

Fatal Crashes vs Age of Drivers

Age of US Drivers and Fatal Crashes

17.7

9.5

6.24.1

2.8 2.4 3.0 3.8

8.0

16.3

0.02.04.06.08.0

10.012.014.016.018.020.0

16 18 20 25 35 45 55 65 75 79

Age of Drivers

Fa

tal

Cra

sh

es

/ 1

00

Mil

lio

n

Mil

es

of

Dri

vin

g

Solution f(x) = 0.013x2 – 1.19x + 28.24 3 = 0.013x2 – 1.19x + 28.24 0.013x2 – 1.19x + 25.24 = 0

a = 0.013; b = -1.19; c = 25.24 -(-1.19) ±√((-1.19)2 – 4(0.013)(25.24))

x = -------------------------------------------- 2(0.013)

x ≈ ((1.19) ± √(0.104)) / (0.26)

x ≈ ((1.19) ± 0.322) / (0.26) x ≈ 33; x ≈ 58

Your Turn Solve the following using the quadratic formula1. 4x2 – 3x = 6

4x2 – 3x - 6 = 0 a = 4; b = -3; c = -6 3 ±√((9 – 4(4)(-6)) 3 ±√(105)

x = ------------------------ = -------------- (2)(4) 8

2. 3x2 = 8x + 7 3x2 - 8x - 7 = 0 a = 3; b = -8; c = -7 8 ±√(64 – 4(3)(-7)) 8 ±√(148) 8 ±√(4 ·37) x = ---------------------------- = -------------- = ----------------- 6 6 6 4 ±√(37) = ------------ 3

Discriminant Given: ax2 + bx + c = 0, where a > 0. -b ± √(b2 – 4ac)

x = ----------------------- 2a

If (b2 – 4ac) >= 0, x are real numbersIf (b2 – 4ac) < 0, x are imaginary numbers.

(b2 – 4ac) is called a discriminant. Thus,

If (b2 – 4ac) >= 0, solution set is real numbers.If (b2 – 4ac) , 0, solution set is complex numbers.

Your Turn

Compute the discriminant and determine the number and type of solutions.

1. x2 + 6x = -9

2. x2 + 2x + 9 = 0

8.3 Quadratic Function & Their Graphs

Quadratic function f(x) = x2 - x - 2

x

f(x)

Characteristics of Quadratic Function Graph f(x) = ax2 + bx + c

Shape is a parabolaIf a> 0, parabola opens upwardIf a < 0, parabola opens downwardVertex is the lowest point (when a > 0), and

the highest point (when a < 0)Axis of symmetry is the line through the

vertex which divides the parabola into two mirror images.

To Sketch a Graph of Quadratic Function Given: f(x) = a(x – h)2 + k Characteristics

If a > 0, parabola opens upwardVertex is at (h, k)If h > 0, graph is shifted to right by h; if h < 0, to

the leftIf k > 0, graph is shifted up by k; if k < 0,

downward by kAxis of symmetry: x = hFor x-intercepts, solve f(x) = 0

Graph of f(x) = -2(x – 3)2 + 8

f(x) = -2(x - 3)^2 + 8

-250

-200

-150

-100

-50

0

50

-7 -6 -5 -4 -3 -2 -1 -0 1 2 3 4 5 6 7 8 9 10 11

x

f(x)

x f(x)

-7 -192

-6 -154

-5 -120

-4 -90

-3 -64

-2 -42

-1 -24

-0 -10

1 0

2 6

3 8

4 6

5 0

6 -10

7 -24

Excel

Graph of f(x) = -2(x – 3)2 + 8 Graph the function: f(x) = -2(x – 3)2 + 8 Solution:

Parabola opens downward (a = -2)Vertex: (3, 8)X-intercepts:

-2(x – 3)2 + 8 = 0(x – 3)2 = -8/-2x – 3 = ± √(4)x = 5; x = 1

y-interceptf(0) = -2(0-3)2 + 8 = -2(9) + 8 = -10

Graph of f(x) = (x + 3)2 + 1

x f(x)

-7 17.0

-6 10.0

-5 5.0

-4 2.0

-3 1.0

-2 2.0

-1 5.0

0 10.0

1 17.0

2 26.0

3 37.0

4 50.0

5 65.0

6 82.0

7 101.0

f(x) = (x + 3)^2 + 1

0.0

20.0

40.0

60.0

80.0

100.0

120.0

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

x

f(x)

Graph of f(x) = (x + 3)2 + 1 Graph the function: f(x) = (x + 3)2 + 1 Solution:

Parabola opens upward (a = 1)Vertex: (-3, 1)X-intercepts:

(x + 3)2 + 1 = 0(x + 3)2 = -1x – 3 = ± ix = 3 + i; x = 3 - i(This means no x-intercepts)

Y-interceptf(0) = (3)2 + 1 = 10

Graphing f(x) = ax2 + bx + c A Solution:

f(x) = a(x2 + (b/a)x) + c = a(x2 + (b/a)x + (b2/4a2)) + c – a(b2/4a2) = a(x + (b/2a))2 + c – b2/(4a)

Comparing to f(x) = a(x – h)2 + k,h = -b/(2a); k = c – b2/(4a)

f(x) = a(x – (-b/(2a))2 + (c – b2/(4a))

Graphing f(x) = 2x2 + 4x - 3 a = 2; b = 4; c = -3 f(x) = a(x – (-b/(2a))2 + (c – b2/(4a))

= 2(x – (-4/(4))2 + (-3 – 16/8) = 2(x + 1)2 – 5

Opens upward Vertex: (-1, -5) X-intercept:

-1 - √(10)/2, -1 + √(10)/2 Y-intercept:

f(0) = -3

Graphing f(x) = 2x2 + 4x - 3

f(x) = 2x^2 + 4x - 3

-20.0

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

x

f(x)