Finite Math Examples

\frac{6}{50 + 23 x^{2} - x^{4}} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{50 + 23 x^{2} - x^{4}} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Factor by grouping.

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Step 1.1.1.1

Reorder terms.

\frac{6}{- x^{4} + 23 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{- x^{4} + 23 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.1.2

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = - 1 \cdot 50 = - 50

$a \cdot c = - 1 \cdot 50 = - 50$ and whose sum is

b = 23

$b = 23$ .

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Step 1.1.1.2.1

Factor

23

$23$ out of

23 x^{2}

$23 x^{2}$ .

\frac{6}{- x^{4} + 23 (x^{2}) + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{- x^{4} + 23 (x^{2}) + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.1.2.2

Rewrite

23

$23$ as

- 2

$- 2$ plus

25

$25$

\frac{6}{- x^{4} + (- 2 + 25) x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{- x^{4} + (- 2 + 25) x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.1.2.3

Apply the distributive property.

\frac{6}{- x^{4} - 2 x^{2} + 25 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{- x^{4} - 2 x^{2} + 25 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

\frac{6}{- x^{4} - 2 x^{2} + 25 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{- x^{4} - 2 x^{2} + 25 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.1.3

Factor out the greatest common factor from each group.

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Step 1.1.1.3.1

Group the first two terms and the last two terms.

\frac{6}{(- x^{4} - 2 x^{2}) + 25 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{(- x^{4} - 2 x^{2}) + 25 x^{2} + 50} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.1.3.2

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

\frac{6}{x^{2} (- x^{2} - 2) - 25 (- x^{2} - 2)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{x^{2} (- x^{2} - 2) - 25 (- x^{2} - 2)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

\frac{6}{x^{2} (- x^{2} - 2) - 25 (- x^{2} - 2)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{x^{2} (- x^{2} - 2) - 25 (- x^{2} - 2)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.1.4

Factor the polynomial by factoring out the greatest common factor,

- x^{2} - 2

$- x^{2} - 2$ .

\frac{6}{(- x^{2} - 2) (x^{2} - 25)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{(- x^{2} - 2) (x^{2} - 25)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

\frac{6}{(- x^{2} - 2) (x^{2} - 25)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{(- x^{2} - 2) (x^{2} - 25)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

\frac{6}{(- x^{2} - 2) (x^{2} - 5^{2})} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{(- x^{2} - 2) (x^{2} - 5^{2})} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.1.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 5

$b = 5$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{3} - 5 x^{2} + 2 x - 10}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Factor out the greatest common factor from each group.

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Step 1.2.1.1

Group the first two terms and the last two terms.

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x^{3} - 5 x^{2}) + 2 x - 10}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x^{3} - 5 x^{2}) + 2 x - 10}$

Step 1.2.1.2

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

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{2} (x - 5) + 2 (x - 5)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{2} (x - 5) + 2 (x - 5)}$

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{2} (x - 5) + 2 (x - 5)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{x^{2} (x - 5) + 2 (x - 5)}$

Step 1.2.2

Factor the polynomial by factoring out the greatest common factor,

x - 5

$x - 5$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (x^{2} + 2)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (x^{2} + 2)}$

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (x^{2} + 2)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (x^{2} + 2)}$

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (x^{2} + 2)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (x^{2} + 2)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor

- 1

$- 1$ out of

x^{2}

$x^{2}$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2}) + 2)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2}) + 2)}$

Step 2.2

Rewrite

2

$2$ as

- 1 (- 2)

$- 1 (- 2)$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2}) - 1 (- 2))}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2}) - 1 (- 2))}$

Step 2.3

Factor

- 1

$- 1$ out of

- 1 (- x^{2}) - 1 (- 2)

$- 1 (- x^{2}) - 1 (- 2)$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}$

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}$

Step 3

To write

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)}$ as a fraction with a common denominator, multiply by

\frac{- 1}{- 1}

$\frac{- 1}{- 1}$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} \cdot \frac{- 1}{- 1} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} \cdot \frac{- 1}{- 1} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}$

Step 4

To write

- \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}

$- \frac{3}{(x - 5) (- 1 (- x^{2} - 2))}$ as a fraction with a common denominator, multiply by

\frac{x + 5}{x + 5}

$\frac{x + 5}{x + 5}$ .

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} \cdot \frac{- 1}{- 1} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))} \cdot \frac{x + 5}{x + 5}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)} \cdot \frac{- 1}{- 1} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))} \cdot \frac{x + 5}{x + 5}$

Step 5

Write each expression with a common denominator of

(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1

$(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 5.1

Multiply

\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)}

$\frac{6}{(- x^{2} - 2) (x + 5) (x - 5)}$ by

\frac{- 1}{- 1}

$\frac{- 1}{- 1}$ .

\frac{6 \cdot - 1}{(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))} \cdot \frac{x + 5}{x + 5}

$\frac{6 \cdot - 1}{(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1} - \frac{3}{(x - 5) (- 1 (- x^{2} - 2))} \cdot \frac{x + 5}{x + 5}$

Step 5.2

Multiply

\frac{3}{(x - 5) (- 1 (- x^{2} - 2))}

$\frac{3}{(x - 5) (- 1 (- x^{2} - 2))}$ by

\frac{x + 5}{x + 5}

$\frac{x + 5}{x + 5}$ .

\frac{6 \cdot - 1}{(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1} - \frac{3 (x + 5)}{(x - 5) (- 1 (- x^{2} - 2)) (x + 5)}

$\frac{6 \cdot - 1}{(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1} - \frac{3 (x + 5)}{(x - 5) (- 1 (- x^{2} - 2)) (x + 5)}$

Step 5.3

Reorder the factors of

(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1

$(- x^{2} - 2) (x + 5) (x - 5) \cdot - 1$ .

\frac{6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)} - \frac{3 (x + 5)}{(x - 5) (- 1 (- x^{2} - 2)) (x + 5)}

$\frac{6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)} - \frac{3 (x + 5)}{(x - 5) (- 1 (- x^{2} - 2)) (x + 5)}$

Step 5.4

Reorder the factors of

(x - 5) (- 1 (- x^{2} - 2)) (x + 5)

$(x - 5) (- 1 (- x^{2} - 2)) (x + 5)$ .

\frac{6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)} - \frac{3 (x + 5)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)} - \frac{3 (x + 5)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

\frac{6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)} - \frac{3 (x + 5)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)} - \frac{3 (x + 5)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 6

Combine the numerators over the common denominator.

\frac{6 \cdot - 1 - 3 (x + 5)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{6 \cdot - 1 - 3 (x + 5)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 7

Simplify the numerator.

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Step 7.1

Factor

- 3

$- 3$ out of

6 \cdot - 1 - 3 (x + 5)

$6 \cdot - 1 - 3 (x + 5)$ .

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Step 7.1.1

Reorder

6 \cdot - 1

$6 \cdot - 1$ and

- 3 (x + 5)

$- 3 (x + 5)$ .

\frac{- 3 (x + 5) + 6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 5) + 6 \cdot - 1}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 7.1.2

Factor

- 3

$- 3$ out of

6 \cdot - 1

$6 \cdot - 1$ .

\frac{- 3 (x + 5) - 3 (- 2 \cdot - 1)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 5) - 3 (- 2 \cdot - 1)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 7.1.3

Factor

- 3

$- 3$ out of

- 3 (x + 5) - 3 (- 2 \cdot - 1)

$- 3 (x + 5) - 3 (- 2 \cdot - 1)$ .

\frac{- 3 (x + 5 - 2 \cdot - 1)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 5 - 2 \cdot - 1)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

\frac{- 3 (x + 5 - 2 \cdot - 1)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 5 - 2 \cdot - 1)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 7.2

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

\frac{- 3 (x + 5 + 2)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 5 + 2)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 7.3

Add

5

$5$ and

2

$2$ .

\frac{- 3 (x + 7)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 7)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

\frac{- 3 (x + 7)}{- (- x^{2} - 2) (x + 5) (x - 5)}

$\frac{- 3 (x + 7)}{- (- x^{2} - 2) (x + 5) (x - 5)}$

Step 8

Simplify terms.

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Step 8.1

Dividing two negative values results in a positive value.

\frac{3 (x + 7)}{((- x^{2} - 2) (x + 5)) (x - 5)}

$\frac{3 (x + 7)}{((- x^{2} - 2) (x + 5)) (x - 5)}$

Step 8.2

Factor

- 1

$- 1$ out of

- x^{2}

$- x^{2}$ .

\frac{3 (x + 7)}{(- (x^{2}) - 2) (x + 5) (x - 5)}

$\frac{3 (x + 7)}{(- (x^{2}) - 2) (x + 5) (x - 5)}$

Step 8.3

Rewrite

- 2

$- 2$ as

- 1 (2)

$- 1 (2)$ .

\frac{3 (x + 7)}{(- (x^{2}) - 1 (2)) (x + 5) (x - 5)}

$\frac{3 (x + 7)}{(- (x^{2}) - 1 (2)) (x + 5) (x - 5)}$

Step 8.4

Factor

- 1

$- 1$ out of

- (x^{2}) - 1 (2)

$- (x^{2}) - 1 (2)$ .

\frac{3 (x + 7)}{- (x^{2} + 2) (x + 5) (x - 5)}

$\frac{3 (x + 7)}{- (x^{2} + 2) (x + 5) (x - 5)}$

Step 8.5

Rewrite negatives.

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Step 8.5.1

Rewrite

- (x^{2} + 2)

$- (x^{2} + 2)$ as

- 1 (x^{2} + 2)

$- 1 (x^{2} + 2)$ .

\frac{3 (x + 7)}{- 1 (x^{2} + 2) (x + 5) (x - 5)}

$\frac{3 (x + 7)}{- 1 (x^{2} + 2) (x + 5) (x - 5)}$

Step 8.5.2

Move the negative in front of the fraction.

- \frac{3 (x + 7)}{((x^{2} + 2) (x + 5)) (x - 5)}

$- \frac{3 (x + 7)}{((x^{2} + 2) (x + 5)) (x - 5)}$

- \frac{3 (x + 7)}{((x^{2} + 2) (x + 5)) (x - 5)}

$- \frac{3 (x + 7)}{((x^{2} + 2) (x + 5)) (x - 5)}$

- \frac{3 (x + 7)}{((x^{2} + 2) (x + 5)) (x - 5)}

$- \frac{3 (x + 7)}{((x^{2} + 2) (x + 5)) (x - 5)}$