How do you solve x^2 / ( x-4) - 7 / (x-4) = 0 and find any extraneous solutions?

Jun 10, 2017

$x = \pm \sqrt{7}$

Explanation:

$\text{since the fractions have a "color(blue)"common denominator}$
$\text{we can combine them by subtracting the numerators}$

$\Rightarrow \frac{{x}^{2} - 7}{x - 4} = 0 \to \left(x \ne 4\right)$

$\text{equating the numerator to zero}$

$\Rightarrow {x}^{2} - 7 = 0$

$\Rightarrow {x}^{2} = 7$

$\textcolor{b l u e}{\text{take the square root of both sides}}$

$\sqrt{{x}^{2}} = \pm \sqrt{7} \leftarrow \text{ note plus or minus}$

$\Rightarrow x = \pm \sqrt{7} \text{ are the solutions}$