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