What is x if 6=7/x+x?

May 3, 2018

Below

Explanation:

$6 = \frac{7}{x} + x$ where $x \ne 0$

$\frac{7}{x} = 6 - x$

${x}^{2} \cdot \frac{7}{x} = {x}^{2} \left(6 - x\right)$

$7 x = 6 {x}^{2} - {x}^{3}$

${x}^{3} - 6 {x}^{2} + 7 x = 0$

$x \left({x}^{2} - 6 x + 7\right) = 0$

$x = 0$ or ${x}^{2} - 6 x + 7 = 0$

For ${x}^{2} - 6 x + 7 = 0$, we need to use the quadratic formula

ie $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{6 \pm \sqrt{36 - 28}}{2}$

$x = \frac{6 \pm 2 \sqrt{2}}{2}$

$x = 3 \pm \sqrt{2}$

BUT looking at $x = 0$, it cannot be a solution because of $\frac{7}{0}$

Therefore, the answer is $x = 3 \pm \sqrt{2}$