Question

How do you solve `x^2 + 10x + 5 = 0` by completing the square?

Answer 1

`x=-5+-2sqrt(5)`
(see below for completing the squares method of solution)

Explanation:

Given:
`color(white)("XXX")x^2+10x+5=0`

Move the constant to the right side as
`color(white)("XXX")color(blue)(x^2+10x)=-5`

We know that `(x+a)^2=color(red)(x^2+2ax+a^2)`
So if the first two terms of a squared binomial are
`color(white)("XXX")color(red)(x^2+2ax) = color(blue)(x^2+10x)`
then
`color(white)("XXX")color(red)(a)=color(blue)(5)`
and we will need to add
`color(white)("XXX")color(red)(a^2)=color(blue)(25)` (to both sides) to complete the square:

`color(white)("XXX")x^2+10x+25 = -5+25`

Writing as a squared binomial and simplifying the right side:
`color(white)("XXX")(x+5)^2=20`

Taking the square root of both sides:
`color(white)("XXX")x+5=+-2sqrt(5)`

Subtracting `5` from both sides
`color(white)("XXX")x=-5+-2sqrt(5)`