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Creating Quadratic Equations By Using Their Roots by pyb17727

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									Creating Quadratic Equations By Using Their Roots

• In our previous work, we’ve been given a quadratic equation
  and asked to find its roots. Using what we know about
  functions, we can also create a quadratic equation by using
  only its roots or zeros. If x = r is a root, then (x − r) is a factor of
  the equation. Therefore, if an equation has roots x = r1 and
   x = r2 , then the resulting equation is (x − r1 )(x − r2 ) = 0 .
• Example:
  Find a quadratic equation with the following roots:
           1
  (a.) − and 3 (b.) 2 − 3 and 2 + 3 (c.) 1 − 2i and 1 + 2i
           2

                                    1
   (a.) Since the roots are x = −     and x = 3 , then the factors are
                                    2
   ⎛    1⎞
   ⎜ x + ⎟ and (x − 3) . Therefore, we have the equation:
   ⎝    2⎠
         ⎛     1⎞
         ⎜ x + 2 ⎟ (x − 3) = 0
         ⎝       ⎠
         (2x + 1)(x − 3) = 0    ← Multiply by 2
   (b.) Since the roots are x = 2 − 3 and x = 2 + 3 , then the
   factors are (x − 2 + 3 ) and (x − 2 − 3 ) . Therefore, we have
   the equation:
          (x − 2 + 3 )(x − 2 − 3 ) = 0
          x 2 − 2x − x 3 − 2x + 4 + 2 3 + x 3 − 2 3 − 3 = 0
          x 2 − 2x − x 3 − 2x + 4 + 2 3 + x 3 − 2 3 − 3 = 0
          x 2 − 2x − 2x + 4 − 3 = 0
          x 2 − 4x + 1 = 0
   (c.) Since the roots are x = 1 − 2i and x = 1 + 2i , then the factors
   are (x − 1 + 2i) and (x − 1 − 2i) . Therefore, we have the equation:
          (x − 1 + 2i)(x − 1 − 2i) = 0
          x 2 − x − 2ix − x + 1 + 2i + 2ix − 2i − 4i2 = 0
          x 2 − x − 2ix − x + 1 + 2i + 2ix − 2i − 4( −1) = 0    ← i2 = −1
          x 2 − 2x + 5 = 0
The Sum and Product of Roots

  • In the previous section, we discovered how to work backwards
    to determine an equation using the given roots. We can also
    create an equation from the sum and product of its roots.
  • Any quadratic equation ax 2 + bx + c = 0 can be expressed in
    the form

          x 2 − (sum of the roots)x + (product of the roots) = 0

     where:

                                                b
                           sum of the roots = −
                                                a
                                                 c
                          product of the roots =
                                                 a

  • Example:
    Calculate the sum and product to determine a quadratic
    equation with roots x = 2 ± 3 .

     Solution:

     The roots are x = 2 − 3 and x = 2 + 3 .

     Therefore, the sum of the roots is:
     (2 − 3 ) + (2 + 3 ) = 4

     The product of the roots is:
     (2 − 3 )(2 + 3 )
     = 4+2 3 −2 3 −3
     =1

     Therefore, the quadratic equation is x 2 − 4x + 1 = 0 .

								
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