Completing the Square (Shortcut)
Completing the Square is where we take a Quadratic Equation
like this ax^2+ bx + c = 0 and turn it into this a(x + d)^2 + e = 0
For those of you in a hurry, I can tell you that:
d = b / 2a
e = c − [ (b^2) / 4a ]
Too short for you to even understand? Let's look at another shortcut but not so short method.
Here is a quick way to get an answer. You may like this method.
First think about the result we want: (x+d)^2 + e
After expanding (x+d)^2 + e we get: x^2 + 2dx + d^2 + e
Now see if we can turn our example into that form to discover d and e
try to fit x^2 + 6x + 7 into x^2 + 2dx + d^2 + e
Now we can "force" an answer:
We know that 6x must end up as 2dx, so d must be 3
Next, we see that 7 must become d^2 + e = 9 + e, so e must be −2
And we get the answer (x+3)^2 − 2
Solving General Quadratic Equations by Completing the Square
We can complete the square to solve a Quadratic Equation (find where it is equal to zero).
But a general Quadratic Equation can have a coefficient of a in front of x^2:
ax^2 + bx + c = 0
But that is easy to deal with ... just divide the whole equation by "a" first, then carry on: