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

1 Answer

Answer:

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.