Basic Math Examples

$\frac{4 v - 1}{v^{2} - 3 v - 28} - \frac{v - 5}{v^{2} - 16}$

Step 1

Simplify each term.

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Step 1.1

Factor $v^{2} - 3 v - 28$ using the AC method.

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Step 1.1.1

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 $- 28$ and whose sum is $- 3$ .

$- 7, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{4 v - 1}{(v - 7) (v + 4)} - \frac{v - 5}{v^{2} - 16}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{4 v - 1}{(v - 7) (v + 4)} - \frac{v - 5}{v^{2} - 4^{2}}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = v$ and $b = 4$ .

$\frac{4 v - 1}{(v - 7) (v + 4)} - \frac{v - 5}{(v + 4) (v - 4)}$

Step 2

To write $\frac{4 v - 1}{(v - 7) (v + 4)}$ as a fraction with a common denominator, multiply by $\frac{v - 4}{v - 4}$ .

$\frac{4 v - 1}{(v - 7) (v + 4)} \cdot \frac{v - 4}{v - 4} - \frac{v - 5}{(v + 4) (v - 4)}$

Step 3

To write $- \frac{v - 5}{(v + 4) (v - 4)}$ as a fraction with a common denominator, multiply by $\frac{v - 7}{v - 7}$ .

$\frac{4 v - 1}{(v - 7) (v + 4)} \cdot \frac{v - 4}{v - 4} - \frac{v - 5}{(v + 4) (v - 4)} \cdot \frac{v - 7}{v - 7}$

Step 4

Write each expression with a common denominator of $(v - 7) (v + 4) (v - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4 v - 1}{(v - 7) (v + 4)}$ by $\frac{v - 4}{v - 4}$ .

$\frac{(4 v - 1) (v - 4)}{(v - 7) (v + 4) (v - 4)} - \frac{v - 5}{(v + 4) (v - 4)} \cdot \frac{v - 7}{v - 7}$

Step 4.2

Multiply $\frac{v - 5}{(v + 4) (v - 4)}$ by $\frac{v - 7}{v - 7}$ .

$\frac{(4 v - 1) (v - 4)}{(v - 7) (v + 4) (v - 4)} - \frac{(v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 4.3

Reorder the factors of $(v - 7) (v + 4) (v - 4)$ .

$\frac{(4 v - 1) (v - 4)}{(v + 4) (v - 4) (v - 7)} - \frac{(v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(4 v - 1) (v - 4) - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(4 v - 1) (v - 4)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{4 v (v - 4) - 1 (v - 4) - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.1.2

Apply the distributive property.

$\frac{4 v \cdot v + 4 v \cdot - 4 - 1 (v - 4) - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.1.3

Apply the distributive property.

$\frac{4 v \cdot v + 4 v \cdot - 4 - 1 v - 1 \cdot - 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $v$ by $v$ by adding the exponents.

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Step 6.2.1.1.1

Move $v$ .

$\frac{4 (v \cdot v) + 4 v \cdot - 4 - 1 v - 1 \cdot - 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.2.1.1.2

Multiply $v$ by $v$ .

$\frac{4 v^{2} + 4 v \cdot - 4 - 1 v - 1 \cdot - 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.2.1.2

Multiply $- 4$ by $4$ .

$\frac{4 v^{2} - 16 v - 1 v - 1 \cdot - 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.2.1.3

Rewrite $- 1 v$ as $- v$ .

$\frac{4 v^{2} - 16 v - v - 1 \cdot - 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.2.1.4

Multiply $- 1$ by $- 4$ .

$\frac{4 v^{2} - 16 v - v + 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.2.2

Subtract $v$ from $- 16 v$ .

$\frac{4 v^{2} - 17 v + 4 - (v - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.3

Apply the distributive property.

$\frac{4 v^{2} - 17 v + 4 + (- v - - 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.4

Multiply $- 1$ by $- 5$ .

$\frac{4 v^{2} - 17 v + 4 + (- v + 5) (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.5

Expand $(- v + 5) (v - 7)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{4 v^{2} - 17 v + 4 - v (v - 7) + 5 (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.5.2

Apply the distributive property.

$\frac{4 v^{2} - 17 v + 4 - v \cdot v - v \cdot - 7 + 5 (v - 7)}{(v + 4) (v - 4) (v - 7)}$

Step 6.5.3

Apply the distributive property.

$\frac{4 v^{2} - 17 v + 4 - v \cdot v - v \cdot - 7 + 5 v + 5 \cdot - 7}{(v + 4) (v - 4) (v - 7)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Multiply $v$ by $v$ by adding the exponents.

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Step 6.6.1.1.1

Move $v$ .

$\frac{4 v^{2} - 17 v + 4 - (v \cdot v) - v \cdot - 7 + 5 v + 5 \cdot - 7}{(v + 4) (v - 4) (v - 7)}$

Step 6.6.1.1.2

Multiply $v$ by $v$ .

$\frac{4 v^{2} - 17 v + 4 - v^{2} - v \cdot - 7 + 5 v + 5 \cdot - 7}{(v + 4) (v - 4) (v - 7)}$

Step 6.6.1.2

Multiply $- 7$ by $- 1$ .

$\frac{4 v^{2} - 17 v + 4 - v^{2} + 7 v + 5 v + 5 \cdot - 7}{(v + 4) (v - 4) (v - 7)}$

Step 6.6.1.3

Multiply $5$ by $- 7$ .

$\frac{4 v^{2} - 17 v + 4 - v^{2} + 7 v + 5 v - 35}{(v + 4) (v - 4) (v - 7)}$

Step 6.6.2

Add $7 v$ and $5 v$ .

$\frac{4 v^{2} - 17 v + 4 - v^{2} + 12 v - 35}{(v + 4) (v - 4) (v - 7)}$

Step 6.7

Subtract $v^{2}$ from $4 v^{2}$ .

$\frac{3 v^{2} - 17 v + 4 + 12 v - 35}{(v + 4) (v - 4) (v - 7)}$

Step 6.8

Add $- 17 v$ and $12 v$ .

$\frac{3 v^{2} - 5 v + 4 - 35}{(v + 4) (v - 4) (v - 7)}$

Step 6.9

Subtract $35$ from $4$ .

$\frac{3 v^{2} - 5 v - 31}{(v + 4) (v - 4) (v - 7)}$