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.
