Completing the Square

Completing the Square is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial .

To solve $a x^{2} + b x + c = 0$ by completing the square:

1. Transform the equation so that the constant term, $c$ , is alone on the right side.
2. If $a$ , the leading coefficient (the coefficient of the $x^{2}$ term), is not equal to $1$ , divide both sides by $a$ .

3. Add the square of half the coefficient of the $x$ -term, ${(\frac{b}{2 a})}^{2}$ to both sides of the equation.

4. Factor the left side as the square of a binomial.

5. Take the square root of both sides. (Remember: ${(x + q)}^{2} = r$ is equivalent to $x + q = \pm \sqrt{r}$ .)

6. Solve for $x$ .

Example 1:

Solve $x^{2} - 6 x - 3 = 0$ by completing the square.

$\begin{array}{l} x^{2} - 6 x = 3 \\ x^{2} - 6 x + {(- 3)}^{2} = 3 + 9 \\ {(x - 3)}^{2} = 12 \\ x - 3 = \pm \sqrt{12} \\ = \pm 2 \sqrt{3} \\ x = 3 \pm 2 \sqrt{3} \end{array}$

Example 2:

Solve: $7 x^{2} - 8 x + 3 = 0$

$\begin{array}{l} 7 x^{2} - 8 x = - 3 \\ x^{2} - \frac{8}{7} x = - \frac{3}{7} \\ x^{2} - \frac{8}{7} x + {(- \frac{4}{7})}^{2} = - \frac{3}{7} + \frac{16}{49} \\ {(x - \frac{4}{7})}^{2} = - \frac{5}{49} \\ x - \frac{4}{7} = \pm \frac{\sqrt{5}}{7} i \\ x = \frac{4}{7} \pm \frac{\sqrt{5}}{7} i \\ {(x - 3)}^{2} = 12 \\ x - 3 = \pm \sqrt{12} \\ = \pm 2 \sqrt{3} \\ x = 3 \pm 2 \sqrt{3} \end{array}$

Completing the Square

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