# Quadratic Functions

### Lauren Sullivan

## I. FINDING SOLUTIONS

## Using Factoring

Find the factors and the x-intercepts of: x² - 7x + 10 = 0

x² - 7x + 10 = 0

**(x - 2) (x - 5) = 0**

**x = 2 & x = 5 **

## Using Square Root Method - Real Solution

(x + 2)² - 16 = 0

+16 +16

(x + 2)² = 16

√(x + 2)² = √16

x + 2 = +- 4

- 2 - 2

**x = -2 +- 4**

## Using Square Root Method - Imaginary Solution

2x² + 16x + 72 = 0

/2 /2 /2

x² + 8x + 36 = 0

-36 -36

x² + 8x ______ = -36 (8 / 2)² = 16

√x² + 8x + √16 = -36

+16 +16

√(x + 4)² = √-20

**x = -4 +- 2i√5**

## Using the Quadratic Formula - Real Solution

ax² + bx + c = 0 x = -b +- √b² - 4ac / 2a

Find the roots by using the quadratic formula: 2x² + 6x = 72x² + 6x = 7

-7 -7

2x² + 6x - 7 = 0

x = -6 +- √36 - (4)(2)(-7) / 4

x = -6 +- √36 + 56 / 4

x = -6 +- √92 / 4

x = -6 +- 2√23 / 4

**x = -3 +- √23 / 2**

## Using the Quadratic Formula - Imaginary Solution

ax² + bx + c = 0 x = -b +- √b² - 4ac / 2a

Find the roots by using the quadratic formula: x² - 6x + 12 = 0x² - 6x + 12 = 0

x = 6 +- √36 - (4)(1)(12) / 2

x = 6 +- √36 - 48 / 2

x = 6 +- √-12 / 2

x = 6 +- 2i√3 / 2

**x = 3 +- i√3**

## By Completing the Square - Real Solution

x² + 6x - 27 = 0

+27 +27

x² + 6x ____ = 27 (6 / 2)² = 9

x² + 6x + 9 = 27

+9 +9

√(x + 3)² = √36

x + 3 = 6

-3 -3

**x = -3 + - 6**

## By Completing the Square - Imaginary Solution

x² + 6x + 14 = 0

-14 -14

x² + 6x _____ = -14 (6 / 2)² = 9

√x² + 6x + √9 = -14

+9 +9

√(x + 3)² = √-5

**x = -3 +- i√5**

## II. DISCRIMINANT

## One Real Solution

ax² + bx + c = 0 b² - 4ac

Find the value of the discriminant and tell the number and nature of the solution:x² - 8x + 16 = 0

8² - (4)(1)(16) = ?

**64 - 64 = 0**

**1 Real Solution**

## Two Real Solutions

ax² + bx + c = 0 b² - 4ac

Find the value of the discriminant and tell the number and nature of the solution:4x² - 12x - 10 = 0

12² - (4)(4)(-10) = ?

**144 - (-160) = 304**

**2 Real Solutions**

## Two Imaginary Solutions

ax² + bx + c = 0 b² - 4ac

Find the value of the discriminant and tell the number and nature of the solution:2x² - 8x + 16 = 0

8² - (4)(2)(16) = ?

**64 - 128 = -64**

**2 Imaginary Solutions**

## III. TYPES OF ANSWERS

## Roots

To find the *roots *we set each factor equal to 0 and solve for x:

x² - x - 6

The factors are: (x - 3) (x + 2)

x - 3 = 0 & x + 2 = 0

**Roots are: x = 3 & x = -2**

## X-Intercepts

*x-intercepts*are the roots, zeros, and solutions.

Example: f(x) = x² - x - 6

f(x) = x² - x - 6

(graph the function and find the x-intercpets)

**X-Intercepts are: x= 3 & x= -2**

## Zeros

*zeros*we set each factor equal to 0 and solve for x:

Example: x² - x - 6

x² - x - 6

The factors are: (x - 3) (x + 2)

x - 3 = 0 & x + 2 = 0

**Roots are: x = 3 & x = -2**

## Solutions

Example: x² - x - 6 = 0

x² - x - 6 = 0

The factors are: (x - 3) (x + 2) = 0

x - 3 = 0 & x + 2 = 0

**Solutions are: x = 3 & x= -2**

## IV. VERTEX FORM OF A QUADRATIC FUNCTION

## With no Number in Front of the Term

x² - 6x - 2 = 0

+2 +2

x² - 6x _____ = 2 (6 / 2)² = 9

√x² - 6x + √9 = 2

+9 +9

(x - 3)² = 11

-11 -11

**f(x) = (x - 3)² - 11**

## With a Number in Front of the Term

-2x² - 4x + 6 = 0

/-2 /-2 /-2

x² + 2x - 3 = 0

+3 +3

x² + 2x _____ = 3 (2 / 2)² = 1

x² + 2x + 1 = 3

+1 +1

√x² + 2x + √1 = 4

(x + 1)² - 4

**f(x) = -2(x + 1)² - 4 **

## V. TRANSFORMATION

## Vertical Shift

**x² + 6**

## Horizontal Shift

**(x - 2)²**

## Vertical Stretch

**-2 (x + 1)² - 4**

## Horizontal Stretch

**- 2/3 (x - 1)² + 8**

## Reflections over the X-Axis

**- (x - 3)² - 11**