# Algebra/Completing the Square

 Algebra ← Factoring Polynomials Completing the Square Quadratic Equation →

## 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 ${\displaystyle y=ax^{2}+bx+c}$ :

1. Divide everything by a, so that the number in front of ${\displaystyle x^{2}}$  is a perfect square (1):

${\displaystyle {\frac {y}{a}}=x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}$

2. Now we want to focus on the term in front of the x. Add the quantity ${\displaystyle \left({\frac {b}{2a}}\right)^{2}}$  to both sides:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$

3. Now notice that on the right, the first three terms factor into a perfect square:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}}$

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$  or, multiplying through by a,

${\displaystyle y=a\left(x+{\frac {b}{2a}}\right)^{2}+c-{\frac {b^{2}}{4a}}}$

## Explanation of Derivation

1. Divide everything by a, so that the number in front of ${\displaystyle x^{2}}$  is a perfect square (1):

${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}={a}}$

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 ${\displaystyle \left({\frac {b}{2a}}\right)^{2}}$  to both sides:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$

3. Now notice that on the right, the first three terms factor into a perfect square:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}}$

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:

${\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}$  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.

 2x2 + 24x + 23 = 0 Does not factor easily, so we complete the square. x2 + 12x + 23/2 = 0 Make coefficient of x2 a 1, by dividing all terms by 2. x2 + 12x = - 23/2 Add – 23/2 to both sides. x2 + 12x + 36 = - 23/2 + 36 Take half of 12 (coefficient of x), and square it. Add to both sides. (x + 6)2 = 49/2 Factor. Now we can take square roots to easily solve this form of the equation. √(x + 6)2 = √49/√2 Take the square root. x + 6 = 7/√2 Simplify. x = -6 + (7√2)/2 Rationalize the denominator.