Algebra Examples

$\frac{v^{2} - 4 v}{5 v^{2} + 9 v - 2} - \frac{2 v + 16}{5 v^{2} + 9 v - 2}$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Combine the numerators over the common denominator.

$\frac{v^{2} - 4 v - (2 v + 16)}{5 v^{2} + 9 v - 2}$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\frac{v^{2} - 4 v - (2 v) - 1 \cdot 16}{5 v^{2} + 9 v - 2}$

Step 1.2.2

Multiply $2$ by $- 1$ .

$\frac{v^{2} - 4 v - 2 v - 1 \cdot 16}{5 v^{2} + 9 v - 2}$

Step 1.2.3

Multiply $- 1$ by $16$ .

$\frac{v^{2} - 4 v - 2 v - 16}{5 v^{2} + 9 v - 2}$

Step 1.3

Subtract $2 v$ from $- 4 v$ .

$\frac{v^{2} - 6 v - 16}{5 v^{2} + 9 v - 2}$

Step 2

Factor $v^{2} - 6 v - 16$ using the AC method.

Tap for more steps...

Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(v - 8) (v + 2)}{5 v^{2} + 9 v - 2}$

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 2 = - 10$ and whose sum is $b = 9$ .

Tap for more steps...

Step 3.1.1

Factor $9$ out of $9 v$ .

$\frac{(v - 8) (v + 2)}{5 v^{2} + 9 (v) - 2}$

Step 3.1.2

Rewrite $9$ as $- 1$ plus $10$

$\frac{(v - 8) (v + 2)}{5 v^{2} + (- 1 + 10) v - 2}$

Step 3.1.3

Apply the distributive property.

$\frac{(v - 8) (v + 2)}{5 v^{2} - 1 v + 10 v - 2}$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(v - 8) (v + 2)}{(5 v^{2} - 1 v) + 10 v - 2}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(v - 8) (v + 2)}{v (5 v - 1) + 2 (5 v - 1)}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $5 v - 1$ .

$\frac{(v - 8) (v + 2)}{(5 v - 1) (v + 2)}$

Step 4

Cancel the common factor of $v + 2$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{(v - 8) (v + 2)}{(5 v - 1) (v + 2)}$

Step 4.2

Rewrite the expression.

$\frac{v - 8}{5 v - 1}$