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
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.