5.6 Solving Quadratic Equations Quadratic Equations are equations of 2nd degree. In other words, they have an x2 term in them. If AxB=0 then A=0 or B=0 This is the basis for the approach we use to solve quadratics in 5.6.

X2 + 5x + 6 = 0 Set = 0 (x + 3)(x + 2) = 0 Factor X+ 3 = 0 OR x + 2 = 0 set each factor = 0 -3 -3 -2 -2 solve for x X = -3 OR x = -2

X2 = 4x + 21 Set = 0 X2 - 4x – 21 = 0 leading coeff + (x - 7)(x + 3) = 0 Factor X - 7 = 0 OR x + 3 = 0 set each factor = 0 X = 7 OR x = -3 solve for x

4X2 = 25 4X2 - 25 = 0 (2x - 5)(2x + 5) = 0 2x - 5 = 0 OR 2x + 5 = 0 2x = 5 OR 2x = -5 X=2 ½ OR x = -2 ½

(x - 2)(x - 1) = 12 x2 - 3x + 2= 12 -12 -12 x2 - 3x – 10 = 0 (x - 5)(x + 2) = 0 x - 5 = 0 OR x + 2 = 0 x = 5 OR x = -2

6.1 Simplifying Rational Expressions Remember what a rational number is? Any number that can be written as a fraction. Rational Expressions then are expressions that are in fraction form. Remember that it is illegal to divide by zero so much like we cannot have zero on the bottom of a numeric fraction, we also cannot have zero on the bottom of an algebraic fraction.

In this rational expression, x=2 To find out what values are excluded for x, set the denominator = 0 and solve for x. This may require you to factor first. When you solve, this tells you what values would make the fraction undefined. These values are exclusions for x. X - 2 = 0 X = 2

there are no exclusions on the x x - 1 = 0 so x=1 x2 – 4 = 0 (x+2)(x-2)=0 so x=2 and x=-2

Just a reminder: a negative fraction can have the negative on top, bottom or out front. BUT not top and bottom. Simplify a rational expression using the quotient rule: 14x2y5z 1y4 28x5yz2 2x3z

Simplifying rational expressions by factoring Factor top and bottom Cancel out any common factors ab – b2 b(a-b) a-b 2b 2b 2 5x3 + 10x2 – 25x 5x(x2 + 2x - 5) (x2+2x-5) 10x2 10x2 2x

X2 + 2x – 3 (x + 3)(x - 1) x - 1 x + 3 x + 3 r2 – 25 (r + 5)(r – 5) r +5 r – 5 r – 5 3x2 – 10x – 8 (3x + 2)(x – 4) 3x +2 X2 +3x – 28 (x + 7)(x – 4) x + 7

Watch out for this! (3x – 7) -(-3x + 7) (7 – 3x)(5 – x) (7 – 3x)(5 – x) -1 1 1 5-x -(5 – x) 5 - x

6.2 Multiplying and Dividing Rational Expressions Remember how to multiply and divide fractions? Mult: top times top; bottom times bottom and reduce Divide: leave the first fraction; flip the second and multiply and then reduce

To multiply or divide rational expressions, you will do the following steps: Factor top and bottom of each fraction Flip if you are dividing Cancel out any common factors Multiply what’s left over

-1