# How do you rationalize the denominator and simplify (x - 42) / (sqrtx+7) - 7?

Jul 3, 2017

See a solution process below:

#### Explanation:

To rationalize the denominator multiply the expression by $\frac{7 - \sqrt{x}}{7 - \sqrt{x}}$

$\frac{7 - \sqrt{x}}{7 - \sqrt{x}} \times \frac{x - 42}{\sqrt{x} + 7} - 7 \implies$

$\frac{7 x - 294 - x \sqrt{x} + 42 \sqrt{x}}{7 \sqrt{x} + 49 - x - 7 \sqrt{x}} - 7 \implies$

$\frac{7 x - 294 - x \sqrt{x} + 42 \sqrt{x}}{49 - x} - 7$

To subtract the $7$ we need to put it over a common denominator:

$\frac{7 x - 294 - x \sqrt{x} + 42 \sqrt{x}}{49 - x} - \left(7 \times \frac{49 - x}{49 - x}\right) \implies$

$\frac{7 x - 294 - x \sqrt{x} + 42 \sqrt{x}}{49 - x} - \frac{343 - 7 x}{49 - x} \implies$

$\frac{7 x - 294 - x \sqrt{x} + 42 \sqrt{x} - 343 + 7 x}{49 - x} \implies$

$\frac{7 x + 7 x - x \sqrt{x} + 42 \sqrt{x} - 294 - 343}{49 - x} \implies$

$\frac{14 x + \left(42 - x\right) \sqrt{x} - 637}{49 - x}$