How do you solve $x^2+x=1$ by completing the square?

Question

6011 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-05-05T13:07:28

$x^2+x=1$

In order for the left side to be a perfect square, it must be the square of some $x+a$ .

Since $(x+a)^2= x^2+2ax+a^2$ ,

we see that we must have $2a=1$ , so $a = 1/2$ .

This make $a^2=1/4$ which I need on the left to make a perfect square. We'll add 1/4 to bot side to get:

$x^2+x+1/4=1+1/4$

Factoring the left and simplifying the right, we get:

$(x+1/2)^2=5/4$ .

Now solve by the square root method:

$x+1/2 = +-sqrt(5/4) = +-sqrt5/2$

So $x=-1/2+-sqrt5/2$ , which may also be written:

$x = (-1+-sqrt5)/2$

How do you solve x^2+x=1x^2+x=1 by completing the square?