# How do you divide \frac { 5a ^ { 2} + 49a - 10} { 14a ^ { 2} + 30a + 4} \div \frac { 5a ^ { 2} - 51a + 10} { 14a ^ { 2} + 30a + 4}?

Jan 16, 2018

$\frac{a + 10}{a - 10}$

#### Explanation:

To begin flip the second fraction upside-down to turn the divide into a multiply:

$\frac{5 {a}^{2} + 49 a - 10}{14 {a}^{2} + 30 a + 4} \div i \mathrm{de} \frac{5 {a}^{2} - 51 a + 10}{14 {a}^{2} + 30 a + 4}$

$= \frac{5 {a}^{2} + 49 a - 10}{14 {a}^{2} + 30 a + 4} \times \frac{14 {a}^{2} + 30 a + 4}{5 {a}^{2} - 51 a + 10}$

Straight away we can cancel the denominator of the first with the numerator of the second like so:

$= \frac{5 {a}^{2} + 49 a - 10}{\cancel{14 {a}^{2} + 30 a + 4}} \times \frac{\cancel{14 {a}^{2} + 30 a + 4}}{5 {a}^{2} - 51 a + 10}$

$= \frac{5 {a}^{2} + 49 a - 10}{1} \times \frac{1}{5 {a}^{2} - 51 a + 10}$

$= \frac{5 {a}^{2} + 49 a - 10}{5 {a}^{2} - 51 a + 10}$

These polynomials can be factorised:

$5 {a}^{2} + 49 a - 10 = \left(5 a - 1\right) \left(a + 10\right)$

and

$5 {a}^{2} - 51 a + 10 = \left(5 a - 1\right) \left(a - 10\right)$

So the fraction now becomes:

$= \frac{5 {a}^{2} + 49 a - 10}{5 {a}^{2} - 51 a + 10} = \frac{\left(5 a - 1\right) \left(a + 10\right)}{\left(5 a - 1\right) \left(a - 10\right)}$

We can now cancel the $5 a - 1$ to leave us with:

$\frac{\cancel{\left(5 a - 1\right)} \left(a + 10\right)}{\cancel{\left(5 a - 1\right)} \left(a - 10\right)} = \frac{a + 10}{a - 10}$