Quadratic Equation by Completing the Square Method
Quadratic Equation by Completing the Square Method
Example 1
$$\textup{Solve } x^2+5x+2=0$$
Solution
\begin{align*}
x^2+5x+2 & = 0\\
x^2+5x & = -2\\
(x)^2+2(x)\left(\frac{5}{2}\right)+\left(\frac{5}{2}\right)^2 & = -2+\left(\frac{5}{2}\right)^2\\
\left(x+\frac{5}{2}\right)^2 & = -2+\frac{25}{4}\\
\left(x+\frac{5}{2}\right)^2 & = \frac{-8+25}{4}\\
\left(x+\frac{5}{2}\right)^2 & = \frac{17}{4}\\
x+\frac{5}{2} & = \pm\sqrt{\frac{17}{4}}\\
x+\frac{5}{2} & = \pm\frac{\sqrt{17}}{2}\\
x & = -\frac{5}{2}\pm\frac{\sqrt{17}}{2}\\
x & = \frac{-5\pm\sqrt{17}}{2}\\
x=\frac{-5+\sqrt{17}}{2}, \,\,\,\, & x=\frac{-5-\sqrt{17}}{2}\\
\end{align*}
Example 2
$$\textup{Solve } x^2+3x-5=0$$
Solution
\begin{align*}
x^2+3x-5 & = 0\\
x^2+3x & = 5\\
(x)^2+2(x)\left(\frac{3}{2}\right)+\left(\frac{3}{2}\right)^2 & = 5+\left(\frac{3}{2}\right)^2\\
\left(x+\frac{3}{2}\right)^2 & = 5+\frac{9}{4}\\
\left(x+\frac{3}{2}\right)^2 & = \frac{20+9}{4}\\
\left(x+\frac{3}{2}\right)^2 & = \frac{29}{4}\\
x+\frac{3}{2} & = \pm\sqrt{\frac{29}{4}}\\
x+\frac{3}{2} & = \pm\frac{\sqrt{29}}{2}\\
x & = -\frac{3}{2}\pm\frac{\sqrt{29}}{2}\\
x & = \frac{-3\pm\sqrt{29}}{2}\\
x=\frac{-3+\sqrt{29}}{2}, \,\,\,\, & x=\frac{-3-\sqrt{29}}{2}\\
\end{align*}
Example 3
$$\textup{Solve } x^2+6x+3=0$$
Solution
\begin{align*}
x^2+6x+3 & = 0\\
x^2+6x & = -3\\
(x)^2+2(x)(3)+(3)^2 & = -3+(3)^2\\
(x+3)^2 & = -3+9\\
(x+3)^2 & = 6\\
x+3 & = \pm\sqrt{6}\\
x & = -3 \pm \sqrt{6}\\
x=-3+\sqrt{6}, \,\,\,\, & x=-3-\sqrt{6}\\
\end{align*}
Example 4
$$\textup{Solve } x^2+8x+4=0$$
Solution
\begin{align*}
x^2+8x+4 & = 0\\
x^2+8x & = -4\\
(x)^2+2(x)(4)+(4)^2 & = -4+(4)^2\\
(x+4)^2 & = -4+16\\
(x+4)^2 & = 12\\
x+4 & = \pm\sqrt{12}\\
x+4 & = \pm\sqrt{4 × 3}\\
x+4 & = \pm\sqrt{4} × \sqrt{3}\\
x+4 & = \pm 2 × \sqrt{3}\\
x+4 & = \pm 2 \sqrt{3}\\
x & = -4 \pm 2 \sqrt{3}\\
x = -4 + 2 \sqrt{3}, \,\,\,\, x & = -4 - 2 \sqrt{3}\\
\end{align*}