# 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

**Example: **

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: