# 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 $a{x}^{2}+bx+c=0$ 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, ${\left(\frac{b}{2a}\right)}^{2}$ 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: ${\left(x+q\right)}^{2}=r$ is equivalent to $x+q=±\sqrt{r}$ .)

6.  Solve for $x$ .

Example 1:

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

$\begin{array}{l}{x}^{2}-6x=3\\ {x}^{2}-6x+{\left(-3\right)}^{2}=3+9\\ {\left(x-3\right)}^{2}=12\\ x-3=±\sqrt{12}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±2\sqrt{3}\\ x=3±2\sqrt{3}\end{array}$

Example 2:

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

$\begin{array}{l}7{x}^{2}-8x=-3\\ {x}^{2}-\frac{8}{7}x=-\frac{3}{7}\\ {x}^{2}-\frac{8}{7}x+{\left(-\frac{4}{7}\right)}^{2}=-\frac{3}{7}+\frac{16}{49}\\ {\left(x-\frac{4}{7}\right)}^{2}=-\frac{5}{49}\\ x-\frac{4}{7}=±\frac{\sqrt{5}}{7}i\\ x=\frac{4}{7}±\frac{\sqrt{5}}{7}i\\ {\left(x-3\right)}^{2}=12\\ x-3=±\sqrt{12}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±2\sqrt{3}\\ x=3±2\sqrt{3}\end{array}$