Solve: [tex]\frac{16}{x^{2}} = \frac{8}{x + 6}[/tex]
[tex]\frac{16}{x^{2}} = \frac{8}{x + 6}[/tex]
[tex]\text{Cross multiply: } 16(x + 6) = 8x^{2}[/tex]
[tex]\text{Simplify: } 2(x + 6) = x^{2}[/tex]
[tex]\text{Expand: } 2x + 12 = x^{2}[/tex]
[tex]\text{Solve for x: } x^{2} - 2x - 12 = 0[/tex]
[tex]\text{Use the quadratic equation: } x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}[/tex]
[tex]x = \frac{2 \pm \sqrt{4 + 48}}{2}[/tex]
[tex]x = \frac{2 \pm \sqrt{52}}{2}[/tex]
[tex]x = \frac{2 \pm \sqrt{13 \cdot 4}}{2}[/tex]
[tex]x = \frac{2 \pm 2\sqrt{13}}{2}[/tex]
[tex]\therefore x = 1 \pm \sqrt{13}[/tex]
