# How do you solve the quadratic equation by completing the square: x^2 - 20x = 0?

Jul 22, 2015

The $\text{-20}$ is the key, half of $\text{-20}$ squared is the number for which you are looking.

#### Explanation:

Completing the square is just like it sounds, you are looking for a number that makes your quadratic a perfect square so it can be written as ${\left(x + a\right)}^{2} = \ldots$

The way to do this is by understanding the distributive property and how it gives you that number that makes a perfect square. So let us write out the expression ${\left(x + a\right)}^{2}$

$\left(x + a\right) \left(x + a\right) = x \cdot x + a \cdot x + a \cdot x + {a}^{s} = {x}^{2} + 2 a x + {a}^{2}$

Notice that there is a $2 a$ and an ${a}^{2}$ in that perfect square.

You should also notice that the $2 a$ is in the same spot as our $\text{-20}$ in the problem we are solving. So how can I find the $a$ that solves this by setting up the equation:

$2 a = - 20 \to a = - 10$

This means that

${a}^{2} = {\left(- 10\right)}^{2} = 100$

So to make ${x}^{2} - 20 x = 0$ a perfect square we need to use $\text{-10}$.

${x}^{2} + 2 \left(- 10\right) x + {\left(- 10\right)}^{2} = {x}^{2} - 20 x + 100$

it looks like we added 100 to make it a perfect square. To keep the equation balanced we need to add 100 to the other side of the equation.

${x}^{2} - 20 x + 100 = 0 + 100$

${x}^{2} - 20 x + 100 = 100$

but now you can factor the quadratic on the left using the $a$ we found above.

${\left(x - 10\right)}^{2} = 100$

and taking the square root of both sides

$\sqrt{{\left(x - 10\right)}^{2}} = \sqrt{100}$

$x - 10 = \pm 10 \implies {x}_{1 , 2} = 10 \pm 10 = \left\{\begin{matrix}{x}_{1} = 10 + 10 = 20 \\ {x}_{2} = 10 - 10 = 0\end{matrix}\right.$

which is what you would have obtained if you had just factored out the greatest common factor ($x$) in the beginning and used the zero product property.

But that is the why and the how of completing the square.