Question

How do you solve `x^2+6x+13=0` by completing the square?

Answer 1

`x=-3+-2i`

Explanation:

Solve by completing the square.

`x^2+6x+13=0`

Move the constant to the right side by subtracting 13 from each side.

`x^2+6xcolor(white)(aaaa)=-13`

Divide the coefficient of the `x` term by `2`.

`x^2+color(red)6xcolor(white)(aaaa)=-13`

`color(red)6/2=color(blue)3`

Square the result and add it to both sides.

`color(blue)3^2=color(magenta)9`

`x^2+6x +color(magenta)9=-13+color(magenta)9`

Factor the left side and simplify the right side.

`(x+color(blue)3)(x+color(blue)3)=-4`

Rewrite as the square of the binomial. Note that the `color(blue)3` in the binomial is the same value `color(blue)3` that resulted from dividing the coefficient of the `x` term by `2`.

`(x+color(blue)3)^2=-4`

Square root both sides and solve for `x`.

`sqrt((x+color(blue)3)^2)=sqrt(-4)`

`x+color(blue)3=+-2i`

`x=-3+-2i`