Basic Math Examples

$\frac{2 a}{a - 7} - \frac{4 a}{a^{2} - 14 a + 49}$

Step 1

Factor using the perfect square rule.

Tap for more steps...

Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{2 a}{a - 7} - \frac{4 a}{a^{2} - 14 a + 7^{2}}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 a = 2 \cdot a \cdot 7$

Step 1.3

Rewrite the polynomial.

$\frac{2 a}{a - 7} - \frac{4 a}{a^{2} - 2 \cdot a \cdot 7 + 7^{2}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 7$ .

$\frac{2 a}{a - 7} - \frac{4 a}{{(a - 7)}^{2}}$

Step 2

To write $\frac{2 a}{a - 7}$ as a fraction with a common denominator, multiply by $\frac{a - 7}{a - 7}$ .

$\frac{2 a}{a - 7} \cdot \frac{a - 7}{a - 7} - \frac{4 a}{{(a - 7)}^{2}}$

Step 3

Write each expression with a common denominator of ${(a - 7)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{2 a}{a - 7}$ by $\frac{a - 7}{a - 7}$ .

$\frac{2 a (a - 7)}{(a - 7) (a - 7)} - \frac{4 a}{{(a - 7)}^{2}}$

Step 3.2

Raise $a - 7$ to the power of $1$ .

$\frac{2 a (a - 7)}{{(a - 7)}^{1} (a - 7)} - \frac{4 a}{{(a - 7)}^{2}}$

Step 3.3

Raise $a - 7$ to the power of $1$ .

$\frac{2 a (a - 7)}{{(a - 7)}^{1} {(a - 7)}^{1}} - \frac{4 a}{{(a - 7)}^{2}}$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 a (a - 7)}{{(a - 7)}^{1 + 1}} - \frac{4 a}{{(a - 7)}^{2}}$

Step 3.5

Add $1$ and $1$ .

$\frac{2 a (a - 7)}{{(a - 7)}^{2}} - \frac{4 a}{{(a - 7)}^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 a (a - 7) - 4 a}{{(a - 7)}^{2}}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Factor $2 a$ out of $2 a (a - 7) - 4 a$ .

Tap for more steps...

Step 5.1.1

Factor $2 a$ out of $- 4 a$ .

$\frac{2 a (a - 7) + 2 a (- 2)}{{(a - 7)}^{2}}$

Step 5.1.2

Factor $2 a$ out of $2 a (a - 7) + 2 a (- 2)$ .

$\frac{2 a (a - 7 - 2)}{{(a - 7)}^{2}}$

Step 5.2

Subtract $2$ from $- 7$ .

$\frac{2 a (a - 9)}{{(a - 7)}^{2}}$