How do you 'complete the square' of a quadratic equation?

To complete the square, we need to rearrange the quadratic equation in the form of ax² + bx + c into the form r(x + p)² + q, where our task is to find the values of the unkowns of r, p and q.  Let's take the example of  completing the square of x² + 3x + 5.  The coeffecient of x² is equal to r, so we can determine early on that the value of r is equal to 1.  The value of p is found by ensuring the coefficient of x is equal to 3.  Since the bracket is being squared, we know that the expanded form of the square bracket will give us x² + 2px + p².  Since 3x and 2px are equal, we can determine that 2p = 3 and therefore p = 3/2.  The last step is to find q.  To find q, we need to subtract the constant formed from the (x + p)² expansion and then add on the constant c that we should have.  Therefore, the value of q is equal to -p² + c.   q = -(3/2)² + 5 = -9/4 + 5 = -9/4 + 20/4 = 11/4.

Therefore, the completed the square form of the quadratic equation x² + 3x + 5 is (x + 3/2)² + 11/4.