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 We will multiply binomials using the FOIL method 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

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 We will factor trinomials 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 )

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 )

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 We will solve quadratic equations by factoring 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

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

We will solve quadratic equations using the quadratic formula      

Using the Quadratic Formula    

Using the Quadratic Formula      

Using the Quadratic Formula  

Application of Quadratics