== Derivation == The purpose of "completing the square" is to either factor a prime quadratic equation or to more easily graph a parabola. The procedure to follow is as follows for a quadratic equation y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} : 1. Divide everything by a, so that the number in front of x 2 {\displaystyle x^{2}} is a perfect square (1): y a = x 2 + b a x + c a {\displaystyle {\frac {y}{a}}=x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}} 2. Now we want to focus on the term in front of the x. Add the quantity ( b 2 a ) 2 {\displaystyle \left({\frac {b}{2a}}\right)^{2}} to both sides: y a + ( b 2 a ) 2 = x 2 + b a x + ( b 2 a ) 2 + c a {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}} 3. Now notice that on the right, the first three terms factor into a perfect square: x 2 + b a x + ( b 2 a ) 2 = ( x + b 2 a ) 2 {\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}} Multiply this back out to convince yourself that this works. 4. Therefore the completed square form of the quadratic is: y a + ( b 2 a ) 2 = ( x + b 2 a ) 2 + c a {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}} or, multiplying through by a, == Explanation of Derivation == 1. Divide everything by a, so that the number in front of x 2 {\displaystyle x^{2}} is a perfect square (1): x 2 + b a x + c a = a {\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}={a}} Think of this as expressing your final result in terms of 1 square x. If your initial equation is 2. Now we want to focus on the term in front of the x. Add the quantity ( b 2 a ) 2 {\displaystyle \left({\frac {b}{2a}}\right)^{2}} to both sides: y a + ( b 2 a ) 2 = x 2 + b a x + ( b 2 a ) 2 + c a {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}} 3. Now notice that on the right, the first three terms factor into a perfect square: x 2 + b a x + ( b 2 a ) 2 = ( x + b 2 a ) 2 {\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}} Multiply this back out to convince yourself that this works. 4. Therefore the completed square form of the quadratic is: y a + ( b 2 a ) 2 = ( x + b 2 a ) 2 + c a {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}} or, multiplying through by a, == Example == The best way to learn to complete a square is through an example. Suppose you want to solve the following equation for x.