Find the centre and radius of the circle with the equation x^2 + y^2 - 8x - 6y - 20 = 0.

The general equation of a circle takes the form (x - A)^2 + (y - B)^2 = r^2 where (A,B) is the centre of the circle with radius r. Thus, we aim to rearrange the equation in the question to match this form. So first we group parts of the equation: x^2 - 8x + y^2 - 6y = 20. Next, we use the completing the square method of factorising. For example, when completing the square of x^2 - 8x we take half the coefficient of x, -4, and obtain a factor of (x - 4)^2. It can be seen when this is expanded out we do obtain x^2 - 8x but we also have a constant of +16. Therefore, for it to balance, we must also include -16. Hence, completing the square of x^2 - 8x obtains (x - 4)^2 - 16. Completing the square for both x and y yields: (x - 4)^2 - 16 + (y - 3)^2 - 9 = 20. We then simplify by adding 9 and 16 to both sides to obtain (x - 4)^2 + (y - 3)^2 = 45 matching the form we were aiming for. Using the definition previously specified, the circle has a centre of (4,3) with a radius of the square root of 45.