1213141516Factoring

Question 1

Find the x-intercepts of

f(x) = 10x² - 8x

Answer 1

Find greatest common factor:

2x(5x - 4)

intercepts are when either piece = 0

x = 0 or ⁴/₅

Question 2

What are the x-intercepts, y-intercept and vertex of:

x² - 11x + 18 = 0

Answer 2

x² - 11x + 18 = 0

x-intercepts: (x - 9)(x - 2) → 9,0 and 9,2

vertex: a = 1; b = -11; c = 18 →

x-vertex = 11 /2 :: y = -12.5

y-intercept = 18

Question 3

What are the x-intercepts, y-intercept and vertex of:

2x² + 8x - 20 = 5x

Answer 3

2x² + 8x - 20 = 5x

2x² + 3x - 20 = 0

(2x - 5)(x + 4) = 0

x-intercepts: (⁵/₂,0) and (-4,0)

y-intercept: (0,-20)

x-vertex: -8/4 = -2

y-vertex: -18

Question 4

14Factor:

3x² - 48 = 0

Answer 4

Create a difference of 2 squares:

3(x² - 16) = 0

3(x - 4)(x + 4) = 0

Question 5

Find roots:

4x² + 36x = 88

Answer 5

4x² + 36x - 88 = 0

4(x² + 9x - 22) = 0

4(x + 11)(x - 2) = 0

roots (x-intercepts) = (-11,0) and (2, 0)

Question 6

What is the square root method for solving quadratic equations?

Answer 6

x² - 36 = 0

x² = 36

x = ±6

Question 7

What is the square root of 36?

Answer 7

It is only 6, not -6.

The square root is always a positive number.

But, with x² - 36 = 0, either +6 or -6 solves it. In this case you are not looking for a square root.

Question 8

Simplify √45

Answer 8

√(9*5)

3√5

Question 9

What is the square root of a negative number?

Answer 9

an imaginary number

a + bi where i = √-1

Question 10

Solve the equation:

x² - 144 = 0

Answer 10

x² - 144 = 0

x² = 144

√x² = ±√144

x = ±12

We are just solving the equation, not finding the square root of x²

Question 11

Solve:

x² - 44 = 20

Answer 11

x² - 44 = 20

x² = 64

√x² = ±√64

x = 8 and -8

Question 12

Solve:

(x - 3)² - 49 = 0

Answer 12

(x - 3)² - 49 = 0

√(x - 3)² = ±√49

x - 3 = ±7

x = 10 or -4

Question 13

Solve:

x² - 63 = 0

Answer 13

x² - 63 = 0

x² = 63

x = ±√63

x = ±3√7

Question 14

Solve:

x² + 48 = 0

Answer 14

x² + 48 = 0

x² = -48

x = ±√(16*3)i

x = ±4√3i

Question 15

What is completing the square?

Answer 15

Manipulating the equation by creating a perfect square.

Question 16

What will it give you?

Answer 16

The vertex.

Question 17

Complete the square and find the vertex:

x² + 6x + 42 = 0

Answer 17

x² + 6x + 42 = 0

^{Divide the 6 by 2 and square it.}

(x² + 6x + 9) + 42 - 9 = 0

(x + 3)² + 33 = 0

vertex-x = -6/2 = -3

:: vertex-y = 33 (plug-in)

y-intercept comes from the original equation → 42

Question 18

What is the vertex of:

y = (x - 5)² + 10

Answer 18

y = (x - 5)² + 10

(5, 10)

When x - 5, (x - 5)² is 0

Question 19

What are the vertex and intercepts of:

y = x² - 10x - 30

Answer 19

y = x² - 10x - 30

y-intercept = -30

y = (x² - 10x + 25) - 30 - 25

y = (x - 5)² - 55

vertex = (5, -55)

a = 1; b = -10; c = -30

^{10±√(100 - - 120)}/₂

5±^√220/₂

x-intercepts: 5±2√55/2 = 5±√55

Question 20

Solve:

y = x² + 7x - 8 = 0

Answer 20

y = x² + 7x - 8 = 0

y = (x² + 7x + ⁷/₂²) - 8 -⁷/₂²

(x + ⁷/₂)² - ⁸¹ /₄

(x + ⁷/₂)² = ± ⁸¹ /₄

x + ⁷/₂ = ±⁹/₂

x = 1 and -8

Question 21

Solve:

-x² + 8x = -40

Answer 21

-x² + 8x = -40

^{faces down}

x² - 8x - 40 = 0

y-intercept = 40

(x² - 8x + 16) - 40 - 16 = 0

(x - 4)² - 56 = 0

vertex = (4, 56)

(x - 4) = ±√56

x-intercepts = (4+2√14,0) and (4-2√14, 0)

Question 22

16Solve:

2x² - 12x - 14 = 0

Answer 22

2x² - 12x - 14 = 0

^{When you make the equation = 0, you are looking for the x-intercepts (where y = 0)}

2(x² - 6x - 7) = 0

^{Divide both sides by 2:}

x² - 6x - 7 = 0

y-intercept = 0

(x² - 6x + 9) - 7 - 9 = 0

(x - 3)² - 16 = 0

vertex = (3, -32) → -16*2

(x - 3) = ±√16

x-intercepts = (-1,0) and (7,0)

Question 23

What does the Quadratic Formula tell you?

Answer 23

It helps you find the roots of a quadratic equation when you can’t do simple factoring.

Question 24

What is the Quadratic formula?

Answer 24

For ax² + bx + c:

^{-b±√(b² - 4ac)}/_2a

Question 25

What is b² - 4ac called?

What does it tell you?

Answer 25

The discriminant.

If it is positive, there are 2 x-intercepts.

If it is 0, there are double roots (it sits on the x-axis)

If it is negative, there are no real roots (it does not cross the x-axis)

Question 26

What does the Quadratic formula tell you about the vertex?

Answer 26

It tells you the x-value of the vertex.

^{-b±√(b² - 4ac)}/_2a

^-b/_2a = x-value vertex

Question 27

How do you solve a Quadratic Inequality?

Answer 27

Graph the = equation and test points inside and out.