Algebra/Completing the Square

Algebra

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 $y=ax^{2}+bx+c$ :

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

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

{\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}+{\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:

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

or, multiplying through by a,

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

Explanation of Derivation

Completing The Square

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

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

{\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}+{\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:

{\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.

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