Algebra/Completing the Square

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 $y=a{x}^2+bx+c$:

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:

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

Multiply this back out to convince yourself that this works.

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

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

Template:TrigBoxOpen $y=a{(x+\frac{b}{2a})}^2+c-\frac{b^2}{4a}$ Template:TrigBoxClose

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.

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

x + 6 = 7/√2

x = -6 + (7√2)/2 Rationalize the denominator.