Completing The Square

Sometimes it’s easier to modify an equation so it becomes a full quadradic equation.

Take the following:

$$x^2+8x+11=0$$

As it is this is not factorable, and while you can use quadratic method instead, you can force it to be factorable!

You can do this whenever the b term is even, this will not work if b term is odd.

First, move the c term to the other side of the equation, then multiply the middle term by 2 and add it to both sides of the equation

$$\begin{array}{rl}{x}^2+8x+16& =-11+16\\ x^2+8x+16& =5\end{array}$$

Now, because the equation is in the format of $x^2+bx+2(b)=c$, we can rewrite it as the following!

$$(x+4{)}^2=5$$

Now you can use the square root property!

$$\begin{array}{rl}\sqrt{(x+4{)}^2}& =\pm \sqrt{5}\\ x+4& =\pm \sqrt{5}\\ x& =-4\pm \sqrt{5}\end{array}$$