Question

How do you solve `x^2 - 8x + 3 = 0` by completing the square?

Answer 1

The solutions are `4 \pm sqrt(13)`.

Explanation:

We must manipulate the expression into something of the form

`x^2 + 2ax+ a^2`

Since the term `2ax` equals `-8x`, it means that `a=-4`, and thus `a^2=16`. We need `13` more units for the constant to reach that value, so we can add and subtract:

`x^2 -8x+ 3 + 13 - 13 = 0 -> x^2 -8x +16 =13`

But now we have that `x^2 -8x +16 = (x-4)^2`, so the equation becomes

`(x-4)^2 = 13`

Taking square roots of both sides:

`x-4 = \pm \sqrt(13)`

`x=4 \pm \sqrt(13)`