Finite Math Examples

$\frac{- 4 x - 3}{x + 5} + \frac{3 x + 4}{x - 5}$

Step 1

To write $\frac{- 4 x - 3}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{- 4 x - 3}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{3 x + 4}{x - 5}$

Step 2

To write $\frac{3 x + 4}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{- 4 x - 3}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{3 x + 4}{x - 5} \cdot \frac{x + 5}{x + 5}$

Step 3

Write each expression with a common denominator of $(x + 5) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{- 4 x - 3}{x + 5}$ by $\frac{x - 5}{x - 5}$ .

$\frac{(- 4 x - 3) (x - 5)}{(x + 5) (x - 5)} + \frac{3 x + 4}{x - 5} \cdot \frac{x + 5}{x + 5}$

Step 3.2

Multiply $\frac{3 x + 4}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(- 4 x - 3) (x - 5)}{(x + 5) (x - 5)} + \frac{(3 x + 4) (x + 5)}{(x - 5) (x + 5)}$

Step 3.3

Reorder the factors of $(x - 5) (x + 5)$ .

$\frac{(- 4 x - 3) (x - 5)}{(x + 5) (x - 5)} + \frac{(3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(- 4 x - 3) (x - 5) + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(- 4 x - 3) (x - 5)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{- 4 x (x - 5) - 3 (x - 5) + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.1.2

Apply the distributive property.

$\frac{- 4 x \cdot x - 4 x \cdot - 5 - 3 (x - 5) + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.1.3

Apply the distributive property.

$\frac{- 4 x \cdot x - 4 x \cdot - 5 - 3 x - 3 \cdot - 5 + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.1.1

Move $x$ .

$\frac{- 4 (x \cdot x) - 4 x \cdot - 5 - 3 x - 3 \cdot - 5 + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.2.1.1.2

Multiply $x$ by $x$ .

$\frac{- 4 x^{2} - 4 x \cdot - 5 - 3 x - 3 \cdot - 5 + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.2.1.2

Multiply $- 5$ by $- 4$ .

$\frac{- 4 x^{2} + 20 x - 3 x - 3 \cdot - 5 + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.2.1.3

Multiply $- 3$ by $- 5$ .

$\frac{- 4 x^{2} + 20 x - 3 x + 15 + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.2.2

Subtract $3 x$ from $20 x$ .

$\frac{- 4 x^{2} + 17 x + 15 + (3 x + 4) (x + 5)}{(x + 5) (x - 5)}$

Step 5.3

Expand $(3 x + 4) (x + 5)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{- 4 x^{2} + 17 x + 15 + 3 x (x + 5) + 4 (x + 5)}{(x + 5) (x - 5)}$

Step 5.3.2

Apply the distributive property.

$\frac{- 4 x^{2} + 17 x + 15 + 3 x \cdot x + 3 x \cdot 5 + 4 (x + 5)}{(x + 5) (x - 5)}$

Step 5.3.3

Apply the distributive property.

$\frac{- 4 x^{2} + 17 x + 15 + 3 x \cdot x + 3 x \cdot 5 + 4 x + 4 \cdot 5}{(x + 5) (x - 5)}$

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.1.1.1

Move $x$ .

$\frac{- 4 x^{2} + 17 x + 15 + 3 (x \cdot x) + 3 x \cdot 5 + 4 x + 4 \cdot 5}{(x + 5) (x - 5)}$

Step 5.4.1.1.2

Multiply $x$ by $x$ .

$\frac{- 4 x^{2} + 17 x + 15 + 3 x^{2} + 3 x \cdot 5 + 4 x + 4 \cdot 5}{(x + 5) (x - 5)}$

Step 5.4.1.2

Multiply $5$ by $3$ .

$\frac{- 4 x^{2} + 17 x + 15 + 3 x^{2} + 15 x + 4 x + 4 \cdot 5}{(x + 5) (x - 5)}$

Step 5.4.1.3

Multiply $4$ by $5$ .

$\frac{- 4 x^{2} + 17 x + 15 + 3 x^{2} + 15 x + 4 x + 20}{(x + 5) (x - 5)}$

Step 5.4.2

Add $15 x$ and $4 x$ .

$\frac{- 4 x^{2} + 17 x + 15 + 3 x^{2} + 19 x + 20}{(x + 5) (x - 5)}$

Step 5.5

Add $- 4 x^{2}$ and $3 x^{2}$ .

$\frac{- x^{2} + 17 x + 15 + 19 x + 20}{(x + 5) (x - 5)}$

Step 5.6

Add $17 x$ and $19 x$ .

$\frac{- x^{2} + 36 x + 15 + 20}{(x + 5) (x - 5)}$

Step 5.7

Add $15$ and $20$ .

$\frac{- x^{2} + 36 x + 35}{(x + 5) (x - 5)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 36 x + 35}{(x + 5) (x - 5)}$

Step 6.2

Factor $- 1$ out of $36 x$ .

$\frac{- (x^{2}) - (- 36 x) + 35}{(x + 5) (x - 5)}$

Step 6.3

Factor $- 1$ out of $- (x^{2}) - (- 36 x)$ .

$\frac{- (x^{2} - 36 x) + 35}{(x + 5) (x - 5)}$

Step 6.4

Rewrite $35$ as $- 1 (- 35)$ .

$\frac{- (x^{2} - 36 x) - 1 (- 35)}{(x + 5) (x - 5)}$

Step 6.5

Factor $- 1$ out of $- (x^{2} - 36 x) - 1 (- 35)$ .

$\frac{- (x^{2} - 36 x - 35)}{(x + 5) (x - 5)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (x^{2} - 36 x - 35)$ as $- 1 (x^{2} - 36 x - 35)$ .

$\frac{- 1 (x^{2} - 36 x - 35)}{(x + 5) (x - 5)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 36 x - 35}{(x + 5) (x - 5)}$