How do you simplify (25-x^2)/12 *(6x^2)/(5-x)?

Apr 27, 2016

$= \frac{{x}^{2} \left(5 + x\right)}{2}$

Explanation:

$\frac{25 - {x}^{2}}{12} \times \frac{6 {x}^{2}}{5 - x}$

Here, $25 - {x}^{2}$ can be written as ${5}^{2} - {x}^{2}$ .

This is of the form ${a}^{2} - {b}^{2}$

And we know that ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Therefore, we have ${5}^{2} - {x}^{2} = \left(5 - x\right) \left(5 + x\right)$

Back to the given expression:

$\frac{25 - {x}^{2}}{12} \times \frac{6 {x}^{2}}{5 - x}$

$= \frac{\left(5 - x\right) \left(5 + x\right)}{12} \times \frac{6 {x}^{2}}{5 - x}$

$\left(5 - x\right)$ is common to the numerator and the denominator, and is cancelled out.

$= \frac{5 + x}{12} \times 6 {x}^{2}$

Next, we know that $2 \times 6 = 12$.

$= \frac{5 + x}{2 \times 6} \times 6 {x}^{2}$

Cancel 6 from the numerator and denominator.

$= \frac{5 + x}{2} \times {x}^{2}$

$= \frac{{x}^{2} \left(5 + x\right)}{2}$