A quadratic equation is one which may be expressed in the following general format:

 

   Ax² + Bx + C = 0

 

   The solution(s) to the quadratic equation is the following:

  

   X = [-B ± √(B² - 4AC)]/2A

 

   x = -B + (B² – 4AC)^1/2  ; x = -B – (B² – 4AC)^1/2
                   2A                              2A

   A few examples:

 

   Solve  2x² – 3x – 2 = 0

 

   First, solve B² – 4AC:   B² – 4AC = (-3)² –4(2)(-2) = 9 + 16 = 25

 

   x = - (-3) +(25)^.5  = 8/4 = 2    ;    x = -(-3) – 5 = -1/2

               2(2)                                        4

A good resource for solving quadratic equations (among other applications) can be found at QuickMath.

 

         1.    x²  -  6x  +  9  =  0

            (x-3)(x-3) = 0

            x =  3; x  =  3

 

       Here, we have two identical solutions to the quadratic solution:  3, 3.

 

       2:             2x²  -  8x  + 6  =  0

                    2(x²  -  4x  +  3)  =  0  (factored out a “2” for simplicity)
                         x²  -  4x  +  3  = 0

                       (x  -  1)(x  -  3) = 0

                                x = 1;  x = 3

 

       Another useful method for solving the quadratic equation is completing the square:

 

3.        Solve  x² + 2x  -  6  =  0

(x² + 2x  +  1)  - 7 =  0    (+1, then –1 to create a perfect square of “x + 1”)

(x+1)²  =  7

x + 1  =  ±√7

x  =  -1  ±√7

 

4.        x²  +  2x  +  6  =  0

(x²  +  2x  +  1) + 6  -1  =  0

(x + 1)²   =  -5

x + 1 =  ±√-5

x = -1  ± i √5     Taking the square root of –5 resulted in two “imaginary solutions.”

      

5.        The “falling body problem”.

 

               If s = constant:       v₀ t ₊  ˝ at²  =  s   

           t =  time (from initial point)       ˝ at²  +  v₀ t   -  s  =  0

                                              t = -v₀  ±  (v₀²  - 4 (.5a)(-1) )^1/2

                                                   2(.5)

                                              t =  -v₀  ±  (v₀² + 2a)^1/2

 

                              

             

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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