Quadratics

Review package

Quadratic Relations

  • One variable is related to another by a function involving constant terms and terms of 1st order or higher.
  • A polynomial with a degree of 2 is Quadratic Relation.
  • For constant increments of the independent variable, a relation is quadratic if the second differences of the dependent variable are constant.
  • A graph that is either concave upwards or concave downwards
  • When the rate of change is constant it is a linear relation
  • When the rate of change is not constant then its not a linear relation

FACTORING

  • Factoring is finding what to multiply to get an expression
  • To factor a number means breaking it into an equation where a number can be multiplied to get the original

There are many different types of factoring I will explain all.

Common Factoring

Firstly we have - Common Factoring


  • If all the terms in the equation are divisible by the same number or variable then that term is called the common factor
  • A polynomial is not completely factored if the GCF (greatest common factor) is not factored out.
  • Expanding and factoring are 2 completely different things a(b+c) <= (factoring) ab+bc <= (Expanding)
Examples



  1. 25x+50
=> 25(x+2)


=> (factored Fully because 25 is the GCF)

2. 25x+50

=> 5(5x+10)

=> (It is NOT factored fully as 5 and 10 still have a common factor)

Grouping

factoring by grouping is factoring in done by grouping pairs of terms. Then factor each pair by groups with ()

Examples

x^2+2x^2+8x+16

=> x^2(x+2) +8(x+2)

=> (x+2) (x^2+8)

SIMPLE TRINOMIALS


  • Simple trinomial equations are in the form of, x^2+bx+c
  • In solving trinomials, the brackets have to be expanded (Multiply each term in the first bracket to each term in the second bracket).
  • Equations in the form of x^2+bx+c are called Quadratic Equations

Expand=> x^2+bx+c , (x+2) (x+1)

  • (x+2) (x+1)
    = x^2+3x+2
  • "c" value is the product. Find out 2 numbers that add up to the "b" value and when multiplied the answer is the "c" value.
Examples:

  1. x^2 +7x -30 [Product=-30 , Sum=7]
    = (x+10) (x-3)


COMPLEX TRINOMIALS

COMPLEX TRINOMIALS

  • Factor by Decomposition.
  • Complex Trinomial are in the form of, Ax^2 +Bx +C.
  • The "C" value is multiplied by the A value, only the coefficient.
  • Determine the factors of the product which also add up to the "B" value.


Factor by Decomposition: Ax^2 +Bx +C


  • 2x^2 +3x -5 [Product: A*C= -5*2=-10 , Sum=3]
    =2x^2 +5x-2x -5 [Decompose the Middle term]
    =x(2x+5) -1(2x+5)
    =(x-1) (2x+5)
Examples:
  1. 5x^2 +10x+2x+4
    =5x(x+2) +2(x+2)
    =(5x+2) (x+2)

FACTORING DIFFERENCE OF SQUARES

FACTORING DIFFERENCE OF SQUARES

  • The "a" value is squared (x^2)
  • In the form of, (a+b) (a-b)
  • Square-root of "b" value

Expand:


  • (x+2) (x-2)


=x^2 -2x+2x -4
=x^2 - 4
  • x^2 -16
    = (x^2 - 4)
Example:

4x^2 - 22
=2(2x^2 - 11)

COMPLETING THE SQUARE

  • Used to change Standard form equations to Vertex form


STEPS TO COMPLETE THE SQUARES:

  1. Block off the first two terms
  2. Factor out the A value only
  3. Divide the middle term by 2 and square it (b/2)^2
  4. Add and subtract the squared value answer
  5. Take out the negative squared value, Multiply the "a" value by the negative squared value when taking it out of the bracket
  6. Write within brackets x + the square root of the number and square the entire bracket and solve what is outside the brackets
Example:

  1. x^2 + 6x - 2
    =(x^2 + 6x) - 2
    =(x^2 + 6x + 9 - 9) -2
    =(x^2 + 6x + 9) -9 - 2
    =(x+3)^2 - 11

QUADRATIC FORMULA

  • In the form of, ax^2 + bx + c = 0
  • Formula is used to identify the "x" value
  • Quadratic formula is used when the equation cannot be factored
  • Quadratic formula give 2 values of "x"
  • Quadratic Formula:

    x = -b ± √b^2 - 4ac /2a

EXAMPLES:

  1. x^2 - x + 8 = 0
    x= -b ± √b^2 - 4ac /2a
    x= 4x^2 + 33x + 8 = 0
    x= -33 ± √33^2 - 4(4)(8) / 2(4)
    x= -33 ± √961 / 8
    x= -33 ± 31 / 8
    x= -1/4 x= -8

THE END HOPE IT HELPED!!!!!!!!!!!!!!
Using the Quadratic Formula
Completing the Square - Solving Quadratic Equations
Factoring difference of squares