# Quadratic Relations

### By: Disani Gnanachandran

## Linear Relations and Quadratic Relations

A quadratic relation is a relation between two variables that can be represented by a quadratic function.The second differences shown in a table will be the same for every row representing a quadratic relation.

## Degree of Polynomials

*Example:*

1) x³-4x+7

Degree=3

**Practice Problems; Find the Degree:**

1)x²(x³-3)

2)a²+bx+c

3)x²+3x-1

## Multiplying Binomials

*Example:*

1) (x+a) (x+b)= x²+bx+ax+ab

= x²+(a+b)x+ab

Practice Problems; Expand and Simplify:

1) (x+2) (x+5)

2) 2(x-3) (x+2)

3) -2(2x+1) (x-4)

## Common Factoring and Factoring by Grouping

*Example:*

*1) ab+ac=a(b+c)*

*2) 4x+20=4(x+5)*

*3) ax-bx-ay+by= (a-b)(x-y)*

**Practice Problems; Factor Completely:**

*1) 8x-12y*

*2) 16abc-12ac+20bc*

*3)12x-16*

**Practice problems; Factor by grouping:**

*1) 3x (x-1)+7 (x-1)*

*2) xy-4y+3x-12*

*3) (x)(2x+1)+(2x+1)*

## Factoring Simple Trinomials

Factoring simple trinomials is finding two terms that multiple to the value of c but add to the value of b as well.

*Example:*

1)x²-5x-14

= (x-7)(x+2)

2)x²+7x+6

= (x+1)(x+6)

**Practice Problems; Factor the following Simple Trinomials:**

1) x²-8x+12

2) x²-4x+4

3) x²-7x-30

## Factoring Complex Trinomials

*Example:*

1) 2x²+3x-5

= (2x+5)(x-1)

## Factoring Difference of Squares

*Example:*

1) x²-y²

=(x+y)(x-y)

Practice Problems; Factor difference of squares:

1) x²-81

2) 18x²-50

3)49x²-25y²

## Completing The Square

- Block off the first 2 terms
- Factor out A
- Add "zero"
- Take the middle term and divide by 2 and square
- and Lastly bring out the negative

*Example:*

1) x²+8x-1

x²+8x+16-16-1

(x+4)(x+4)-17

(x+4)²-17

Practice Problems:

1) x²+24x-8

2) 3x²+24x-11

3) -5x²+20x-2

## Quadratic Formula

1) 4x²+33x+8=0

2) -2x²+3x+11=0

3) 6x²-x-15=0