Question

How do you solve by completing the square: `x^2 - 4x + 2 = 0`?

Answer 1

We know that `(x+a)^2 = x^2+2ax+a^2`. Since your equation starts with `x^2-4x`, it means that we must find a value of `a` such that `2ax=-4x`, which means `a=-2`.

If `a=-2`, then we have `(x-2)^2 = x^2-4x+4`. You can see that your equation `x^2-4x+2` is very close: we only need to add `2`.

So, adding `2` to both sides, we get `x^2-4x+4=2`, which means

`(x-2)^2 = 2`. Extracting square roots at both sides, we get `x-2= \pm\sqrt(2)`, which leads us to the solutions

`x_{1,2} = 2 \pm \sqrt(2)`