We begin with x^2-14x=0. I like to deal with equations in standard form (color(red)(a)color(green)(x)^2+color(blue)(b)color(green)(x)+color(orange)(c)), so I'll just rewrite this to x^2-14x+0=0. From here, we need to solve for x, and we need to do that by completing the square. Completing the square is a method where we take an equation that is not a perfect square and find a value that could make it factorable.
For us, our first step is to make sure that the color(red)(a) in color(red)(a)color(green)(x)^2+color(blue)(b)color(green)(x)+color(orange)(c) is a 1. Is that the case for x^2-14x+0=0? Yes, it is.
Now we take the second coefficient (color(blue)(b)), and divide it in half (-14/2), in our case giving us -7. We then take that value and square it, giving us 49 (-7^2). We take that number and add it to our equation, like this:
x^2-14x+49+0=0.
WAIT!!!! We just added a random number into this equation! We can't do that. We need to keep the equation equal. We could add a 49 on the other side, or we could just subtract the 49. Then the numbers cancel each other out and don't change the value of the equation.
NOW we have x^2-14x+49-49+0=0. Now, we went to a lot of effort to get x^2-14x+49, because this is a perfect square. We can rewrite that part of the equation into (x-7)^2. We still need to deal with -49+0, so we just combine them to give us -49. Put it together and we get (x-7)^2-49=0.
Now we just need to solve for x. I'll go through this part quickly.
(x-7)^2-49=0
(x-7)^2=49
sqrt((x-7)^2)=sqrt(49)
x-7=+-7
x=7+-7
x=0, 14
