6.6 Quadratic Equations-We will multiply binomials using the FOIL method.

-We will factor trinomials

-We will solve quadratic equations by factoring.

-We will solve quadratic equations using the quadratic formula

-We will solve problems modeled by quadratic equations.

Multiplying binomials using FOIL

• Binomial - a simplified algebraic expression that contains two terms in which each exponent that appears on a variable is a whole number.

• X+3 y-7 3x2 - 3x

• FOIL - First, Outside, Inside, Last

We will multiply binomials using the FOIL method

Using FOIL

• (ax + b)(cx + d) = ax.cx+ ax.d + b.cx + b.d

• ax.cx = First

• ax.d = Outside

• b.cx = Inside

• b.d = Last

Using Foil with numbers

• (8-3)(4+8) We know = 5 . 12 = 60

• Using Foil:

• 8 . 4 = 32

• 8 . 8 = 64

• -3 . 4 = -12

• -3 . 8 = -24

• 32 + 64 - 12 - 24 = 60

Using FOIL for Binomials

• (x + 8)(x-3) =

• First x . x = x2

• Outside x . -3 = -3x

• Inside 8 . x = 8x

• Last 8 . -3 = -24

• Put it together x2 + 5x - 24

Using FOIL for Binomials

• (3x + 2)(4x-5) =

• First 3x . 4x = 12x2

• Outside 3x . -10 = -30x

• Inside 2 . 4x = 8x

• Last 2 . -5 = -10

• Put it together 12x2 -30x - 10

Factoring a trinomial where the coefficient of the squared term is 1• Let take x2 + 10x +24. • We need to think of FOIL in Reverse• First we know the factors are:

– (x+ )(x+ )

– We know the plus signs because the coefficient of x is positive and because 24 is positive both signs are the same.

• Now we need to think of factors of 24 that when added together equal 10. 6, 4– ( x + 6 )( x + 4 )

We will factor trinomials

Factoring Trinomials with x2

• If a trinomial can not be factored it is considered to be prime

• x2 + 6x + 9 x2 + 17 x + 72

Factoring a Trinomial where a≠1

• 8x2 + 16x - 24

• Find two terms whose product is 8x2

• List all the factors of -24

• Try all the combinations of these factors

• Verify the factorization using the FOIL method

Working with 5x2 + 7x - 24

• (x )(5x )

• (x )(5x )

• (x )(5x )

• (x )(5x )

• (x )(5x )

• (x )(5x )

Factoring Trinomials with x2

• x2 + x - 12 x2 + 21 x - 72

Factoring Trinomials with x2

• x2 - 7x + 12 x2 - 27 x + 72

Solving a Quadratic Equation by Factoring

• A quadratic equation - Any equation that can be written in the form ax2 + bx + c = 0– Where a, b and c are real numbers a ≠ 0

• Zero Product Principle– If AB = 0, then A = 0 or B = 0

Example: x2 + 5x + 6 = 0

We will solve quadratic equations by factoring

Solving a factored quadratic equation

• (x - 5)(x + 2) = 0– Either x - 5 = 0 or x + 2 = 0– X = {5, -2}– Check your results

• Also this is x2 - 3x - 10 = 0– Check x = {5, -2} in the quadratic form.

Solving a quadratic equation by factoring

• x2 + 7x = 18 x2 = 5x - 36

Solving a quadratic equation by factoring

• x2 -16x = -3x2 - 15 7x2 = -13x + 24

The Quadratic Formula

-We will solve quadratic equations using the quadratic formula

Using the Quadratic Formula

Using the Quadratic Formula

Using the Quadratic Formula

Application of Quadratics