COMPLETING THE SQUARE
The QUADRATIC FORMULA (hidden behind a secret panel) 
Suppose you need to solve this grim looking quadratic equation (and we're warning you, the answers are not whole numbers!):  x^{2} + 5x  9 = 0 
First move the constant across. Here we do it by adding 9 to both sides:  x^{2} + 5x = 9 
Now we're going to make a new equation by playing with the Left Hand Side (or LHS). We'll ignore the RHS for a moment. Follow these instructions exactly.  
1/ divide the LHS by x . In this case we get  x + 5 
2/ divide the number by 2 (don't divide the "x"). Now we get  x +2.5 
3/ put both terms in a bracket, square it and then multiply it all out 
(x + 2.5)^{2}
= (x +2.5)(x +2.5) = x^{2} + 2.5x + 2.5x + 6.25 = x^{2} + 5x + 6.25 
Here comes the coolest part of the whole operation. Because we know that x^{2} + 5x =9 (look back a few lines , you'll find it written there!) we can swap the x^{2} +5x on the RHS for 9.  
Now we've got our new equation  (x +2.5)^{2} = 9 + 6.25 
Then a quick little sum gives us...  (x + 2·5)^{2} = 15·25 
We now take the square root of both sides, and the clever bit is that square roots can be + or   
Here we get: 
x + 2·5 = + sqrt(15.25)
OR x + 2.5 =  sqrt(15.25) 
Grab a calculator to work out the square root... 
x + 2·5 = + 3.905
OR x + 2.5 =  3.905 
And then when you take away the 2·5 from both sides you get the two solutions 
x = +3·905 2.5 = 1·405
OR x = 3·905 2.5 =  6·405 

Here's another one just to make sure you've got it. To make it more exciting this one has an x^{2} coefficient!
We're going to solve this little baby...  3x^{2}  11x  8 = 0 
Move the constant over  3x^{2}  11x = 8 
Before we go on, we divide everything by the x^{2} coefficient because we want the x^{2} by itself  x^{2}  11x/3 = 8/3 
Here's where we start making our new equation...  
Make the LHS into a square using steps 1,2 and 3 as before. (So divide through by x, then divide the constant by 2, then put the answer in a bracket and square it.)  (x  11/6)^{2} 
Multiply the square out 
(x  11/6)^{2} = x^{2}  11x/6  11x/6 +121/36
= x^{2}  11x/3 + 121/36 
Now it's time to play the cool little trick! We know from before that x^{2}  11x/3 = 8/3 so we put this into the RHS...  
... and here's the new equation:  (x  11/6)^{2} = 8/3 + 121/36 
At this point we'll make everything into decimals:  (x  1.833)^{2} = 2.667 + 3.361 = 6.028 
Take square roots of both sides 
x  1.833 = + sqrt(6.028)
OR x  1.833 =  sqrt(6.028) 
Get the calculator and work out the square root... 
x  1.833 = + 2.455
OR x  1.833 =  2.455 
If we add 1.833 to both sides we get you get the two answers: 
x = +1·833 + 2·455 = 4·288
OR x = +1·833 2.455 =  0·622 