More Solving Quadratic Equations:
The Quadratic Formula

Again, what we've all been waiting for.

The quadratic formula is pretty easy to remember, and will always work, even if the solutions to the quadratic equation aren't always real numbers (they can be imaginary - the square roots of negative numbers).  So here it is:

                _______
x = -b ± √b² - 4ac
                2a

First, just try it.  Then we will do some more work with it.  Take the equation 2x² + 7x + 6 = 0.

              __________
x = -7±√7² - 4(2)(6)
                2(2)

x = -7 ± √1
         4

x = -8/4 = -2 OR x = -6/4 = -1.5

Try x² - 2x + 4 = 0.

                  ___________
x = -(-2)±√(-2)² - 4(1)(4)
                    2(1)

            ___
x = 4±√-12
           2

As you can see, this equation has no real solutions, because we need to take the square root of a negative number.

Now that you have seen these two examples, it is time to see how the quadratic formula works, and it works on the basis of the "completing the square" process.

So we have the standard form of the equation: ax² + bx + c = 0

Divide each term by a, to get x² alone (you need this for completing the square):
x² +  b x +  c  = 0
         a        a

Then move the constant, c/a, to the other side by subtracting it from both sides:
x² +  b x = - c 
         a         a

Now on the left side we want to complete the square.  Remember that that form is a² + 2ab + b².  Basically, in completing the square, to get the term of b², when the coefficient of x² is 1, we just add the square of half the coefficient of x.  Ok...in a simpler showing:
 1  *  b  =  b 
 2      a     2a

( b ² =  b²  
2a)       4a²

Add this ^^ to both sides.

x² +  b x +  b²   = - c  +  b²  
         a        4a²       a      4a²

We factor the left side and rearrange the right side.

(x +  b )² =  b²   -  c 
          2a       4a²      a

Now get a common denominator for the right side.  It will be 4a².

(x +  b )² =  b²   -  4ac 
          2a       4a²      4a²

(x +  b )² =  b²   -  4ac 
          2a               4a²

Take the square root of each side, but remember on the right side to put plus OR minus.
                     _________
  x +  b  = ±√ b²   -  4ac 
          2a          4a²        

(I can't show it well, but on the right side, it's the square root of the WHOLE SIDE, not just the top line.  Now simplify the right side again.
                     ________
  x +  b  = ±√ b²   -  4ac 
          2a         √4a²    
                     ________
  x +  b  = ±√ b²   -  4ac 
          2a         2a

We're almost done!  Subtract b/2a from both sides, to move it to the right.
                     ________
  x = -  b  ± √ b²   -  4ac 
            2a         2a  

Here's the last step:  Simplify the right side again.
                    ________
 x = -  b  ± √ b²   -  4ac
                       2a

Recognize this???  Well you should...*gives people who don't glares*...just kidding.  It really takes awhile to get the hang of all this.  You may have to go over this explanation more than a few times to get it.  You should know that it is easiest to factor a quadratic, if possible, but this can also be used.  Happy quadratic equation solving!!

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