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					Solving Equations by
Factoring


Module 8, Lesson 6
Online Algebra
VHS@PWCS
         Solving Quadratic Equations

So far we have solved quadratic equations by:
1. Taking the square root of both sides.
    We can do this when the equation is of the form ax2
       + c = 0.
        Note that the only variable in this type is squared.

2. The quadratic formula.
    We can use this to solve any quadratic equations.

                         b  b 2  4ac
                      x
                              2a
Solve the following equations

        x2 = 25                x2 - 63 = 81

Take the square root of     Get the x2 by itself.
   both sides.            x2 – 63 + 63 = 81 + 63
         x=5                     x2 = 144
         x = -5
                          Take the square root of
     Remember that               both sides.
  quadratics can have 2           x = 12
       solutions!
                                  x = -12
             Solve the following equation.
             Use the quadratic formula.
              2y2 + 3 = -7y                      7  7 2  4(2)(3)
                                              x
                                                      2(2)
1.   Get everything on the same side.
                                                   7  49  24
           2y2 + 7y + 3 = -7y + 7y              x
                                                         4
               2y2 + 7y + 3 = 0
                                                     7  25
2.   Identify a, b and c.                         x
                                                         4
            a = 2, b = 7 and c = 3
                                                       7  5
3.   Substitute into the quadratic formula.        x
                                                         4
             b  b 2  4ac                          x  3
          x                                              1
                  2a                                 x
                                                          2
     Zero Product Property

The Zero Product Property states that:
  For all numbers a and b, if ab = 0, then a = 0,
  b= 0 or both a and b equal 0.

This makes sense since 0 times any number will
  always give you zero.

We are going to use this property to solve
 quadratic equations by factoring.
        Solve the following equation.
                    (y + 2)(3y + 5) = 0
 According to the zero product property either (y + 2) or
  (3y + 5) must equal zero.
 We will assume that each equals zero and set them
  equal to zero.
             y+2=0                       3y + 5 = 0
 Then we will solve.
      y + 2 – 2 = 0 + -2            3y + 5 + -5 = 0 + -5
             y = -2                        3y = -5
                                             y = -5/3
Solve the following equation.

                   y(y – 12) = 0

1. Set each factor equal to zero.
   y=0                     y – 12 = 0
2. Solve.
   y=0              y – 12 + 12 = 0 + 12
                              y = 12.

   Remember quadratics can have 2 solutions.
        Solving equations by factoring
 Sometimes we need to factor an equation before setting it
                        equal to zero.
                   Solve m2 + 36m = 0
1. Factor. Here we can pull out the GCF of m.
                     m(m + 36) = 0
2. Set each factor equal to zero.
     m=0                          m + 36 = 0
3. Solve.
     m=0                   m + 36 + -36 = 0 + -36
                                      m = -36
4. We have 2 solutions m = 0 and m = -36.
        Solve: a2 + 4a = 21

1. Just like when using the             a2 + 4a = 21
   quadratic formula, we need     a2 + 4a + -21 = 21 + -21
   one side to be zero.               a2 + 4a – 21 = 0
2. Factor
3. Set both factors equal to
                                         (a + 7)(a – 3) = 0
   zero.
4. Solve
                                a+7=0            a–3=0
                                                  a=3
                                a = -7
           Try these! Click for the
           answers.        c – 17c + 60 = 0 2

                                           (c -12)(c – 5) = 0
1. c2 – 17c + 60 = 0             c – 12 = 0           c–5=0
                                     c = 12              c=5
2. a2 + 64 = -16a
                                            a2 + 64 = -16a
                                      a2 + 64 + 16 a = -16a + 16a
Since these factors are the same
                                           a2 + 16a + 64 = 0
we only need to set one equal to
zero. This quadratic will have only        (a + 8)(a + 8) = 0
one solution. a = -8
                                                a+8=0
                                                a = -8
       Solving Quadratics

We now have 3 different ways to solve quadratic
  equations.
 Taking the square root of both sides.
     This only works when the equation is of the form
      ax2 + c = 0.
 The quadratic formula
     This can be used with any quadratic equation.
 By factoring and setting factors equal to zero.

				
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