Question

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

Answer 1

The `"-20"` is the key, half of `"-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 `(x+a)^2 = ...`

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 `(x+a)^2`

`(x+a)(x+a) = x*x+a*x+a*x+a^s = x^2 + 2ax + a^2`

Notice that there is a `2a` and an `a^2` in that perfect square.

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

`2a = -20 -> a = -10`

This means that

`a^2 = (-10)^2 = 100`

So to make `x^2-20x = 0` a perfect square we need to use `"-10"`.

`x^2 + 2(-10)x + (-10)^2 = x^2 -20x + 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 - 20x +100 = 0 + 100`

`x^2 -20x + 100 = 100`

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

`(x-10)^2 = 100`

and taking the square root of both sides

`sqrt((x-10)^2) = sqrt(100)`

`x-10 = +-10 implies x_(1,2) = 10 +- 10 = {(x_1 = 10 + 10 = 20), (x_2 = 10 - 10 = 0) :}`

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.