Theory

Complete the square

To complete the square for an expression in the form

x^{2} + b x,

the term

c = (\frac{b}{2})^{2}

must be added. To maintain equivalence with the original expression,

c

should also be subtracted.

x^{2} + b x ⇕ x^{2} + b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2}

Putting the subtracted

c

aside temporarily, the expression can be rewritten as a perfect square.

x^{2} + b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2} ⇕ (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2}

To complete the square for an expression in the form

x^{2} - b x,

the method is similar.

x^{2} - b x ⇕ x^{2} - b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2}

Putting the subtracted

c

aside temporarily, the expression can be rewritten as a perfect square.

x^{2} - b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2} ⇕ (x - \frac{b}{2})^{2} - (\frac{b}{2})^{2}