Solve the following equation by completing the square: x^2 + 6x + 3 = 0.

Completing the square is a method of solving quadratic equations that cannot be easily factorised, without having to use the quadratic formula. The first step is to look at the coefficient of the second term: in the above question, this is 6. We then halve the coefficient, i.e. 3. We can then add this to x and square the whole term, as below:

We have (x + 3)². This equals x² + 6x + 9.
So we have the right x² and x terms, but not the right constant.
To make this equal the above equation, we need to subtract 6 and equate to 0. So:
x² + 6x + 3 = (x + 3)² - 6 = 0.
We have completed the square!

We can then solve the equation
(x + 3)² - 6 = 0
(x + 3)² = 6
x + 3 = +/- rt(6)
x = -3 +/- rt(6)