Completing the Square

Converting from Quadratic to Parabolic form

The purpose of completing the square is to convert an equation from quadratic form (y = ax2 + bx + c) to parabolic form (y = a(x - h)2 + k). The first step is to factor "a" from the first two terms. The second step is to find the perfect square term. The third term of a perfect square is always the square of 1/2 the coefficient of the "x" term. Hence, 1/2 of b/a is b/(2a). Squaring this value, the term which must be added to complete the square is (b2)/(4a2). However, if we add a term to the side, we must balance it by subtracting the same value from the side. Using the distributive rule, a * (b2)/(4a2) is (b2)/(4a) which is the value of what was added to the right side of the equation. To balance the equation, we must subtract the same value. The third step is to re-write the equation as a perfect square. Factoring the quadratic portion of the equation, one finds that x2 + (b/a) x + b2/(4a2) factors to [x + b/(2a)]2. The equation is now in parabolic form of y = a(x-h)2 + k. "h" is equal to -b/(2a) and k is equal to c2 - b2/(4a).

Return to Linearizing the Ball Lab.

Return to Linearizing the Fence Lab.

Return to Acceleration down the Ramp Lab