Quadratic Relations

By: Disani Gnanachandran

Linear Relations and Quadratic Relations

A linear relation is a relation between two variables that appears as a straight line when it is graphed. The first differences shown in a table will be the same for every row in the table.


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.

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Degree of Polynomials

A polynomial is an algebraic expression consisting of one or more terms. The degree of a one variable polynomial is the greatest exponent that appears in any term of the expanded form of the polynomial.


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

A binomial is an algebraic expression that contains two terms, for example 4x-7y. The distributive property is an algebraic property which is used to multiply a single term and two or more terms inside a parentheses. Using the distributive property expand the binomials and collect terms to simplify. You can also use the "FOIL" method.


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)

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Common Factoring and Factoring by Grouping

Factoring is the opposite of expanding. If every term is divisible by the same greatest constant, the constant is called a common factor.


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

Form: x²+bx+c into (x+/-_)(x+/-_)

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 Trinomials - MathHelp.com - Algebra Help

Factoring Complex Trinomials

Factoring complex trinomials is multiplying term A and C together and finding the two terms that multiple to that value but add to the value of B as well.


Example:

1) 2x²+3x-5

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

Factoring Complex Trinomials

Factoring Difference of Squares

Factoring the difference of squares is making sure the question can be square rooted.


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

Completing the square is changing the equation from standard from into vertex form. To complete the square:


  1. Block off the first 2 terms
  2. Factor out A
  3. Add "zero"
  4. Take the middle term and divide by 2 and square
  5. 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

Practice Problems:

1) 4x²+33x+8=0

2) -2x²+3x+11=0

3) 6x²-x-15=0

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Algebra Help - The Quadratic Formula - MathHelp.com

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