Quadratics

Review Package

Quadratic Relations

  • Sometimes a curve of best fit is more appropriate model for data than a line of best fit.
  • For constant increments of the independent variable (x), a relation is quadratic if the second differences of the dependent variable (y) constant.
    Example: In a table of values the first differences are not constant but the second differences are constant.
  • A polynomial of degree 2 models a quadratic relation.
  • A linear relation models a phenomenon where the rate of change is constant.
  • A non-linear relation models a phenomenon where variable is the rate of change.

Factoring

COMMON FACTOR

  • Factoring is the opposite of expanding, ab+ac = a(b+c)
  • If every term of a polynomial is divisible by the same constant, the constant is called a common factor.
  • A polynomial is not considered to be completely factored until the Great Common Factor (GCF) has been factored out.

Factor: ab+ac
= a(b+c) , 4x+20
= (x+5)

  • 4x+20
    = 2(2x+10) (Not completely Factored)
  • 4x+20
    = 4(x+5) (Completely Factored)


EXAMPLES:

  1. 3x-15y
    = 3(x+5y)
  2. 4a^3b^4 - 6a^2b^2 + 2ab
    = 2ab(2a^2b^3 - 3ab+1)

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FACTOR BY GROUPING

  • This is done by grouping a pair of terms. Then, factor each pair of two terms.
  • x^2 + 2x^2 + 8x+16
    = x^2(x+2) +8(x+2)
    = (x+2) (x^2+8)
  • 8am-3bn-6an+4bm
    = 4m(2a-b) -3n(b+2a)

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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 +6 [Product=6 , Sum=7]
    = (x+1) (x+6)
  2. x^2 +7x -30 [Product=-30 , Sum=7]
    = (x+10) (x-3)
  3. x^2 +3x +5 [Product=5 , Sum=3]
    =Not Factor-able


*Note: In some cases, first you might have to factor out.
Example:

Q: -5x^4 +5x^3 +30x^2 [First factor and then solve as a Simple Trinomial]
=-5x^2(x^2-x-6) [Product=-6 , Sum=-1]

=-5x^2(x-3)(x+2)

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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. 14x^2 -19x -3 [Product: A*C= 14*-3=-42 , Sum=-19]
    =5x^2 +10x+2x +4 [Decompose the Middle term]
    =5x(x+2) +2(x+2)
    =(5x+2) (x+2)
  2. x^2 +7x +12 [Product: A*C= 12*1=12 , Sum=7]
    =x^2 +3x+4x +12 [Decompose the Middle term]
    =x(x+3) +4(x+3)
    =(x+1) (x+3)
  3. 5x^2 +10x+2x+4
    =5x(x+2) +2(x+2)
    =(5x+2) (x+2)

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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)


EXAMPLES:

  1. (a-3) (a+3)
    =a^2 +3a-3a -9
    =a^2 - 9
  2. (p-5) (p+5)
    =p^2 - 25
  3. 4x^2 - 22
    =2(2x^2 - 11)

Factoring binomials - difference of squares: Quick Explanation!

COMPLETING THE SQUARES

  • 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


EXAMPLES:


  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
  2. 3x^2 + 24x - 11
    =3(x^2 + 8x) - 11
    =3(x^2 + 8x + 16 -16) - 11
    =3(x^2 + 8x +16) -48 -11
    =3(x^2+4)^2 - 59

Completing the Square to Solve Quadratic Equations: More Examples - 1

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
  2. 24= x^2 - x +30
    x^2 - x + 6 = 0
    x= -b ± √b^2 - 4ac /2a
    x= -(-1) ± √(-)^2 - 4(1)(6) / 2(1)
    x= 1± √-23 / 2
    x= No Solution