Finite Math Examples

\frac{x^{2} - 48}{x^{2} + 15 x + 56} - \frac{x - 8}{x + 8}

$\frac{x^{2} - 48}{x^{2} + 15 x + 56} - \frac{x - 8}{x + 8}$

Step 1

Factor

x^{2} + 15 x + 56

$x^{2} + 15 x + 56$ using the AC method.

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Step 1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

56

$56$ and whose sum is

15

$15$ .

7, 8

$7, 8$

Step 1.2

Write the factored form using these integers.

\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{x - 8}{x + 8}

$\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{x - 8}{x + 8}$

\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{x - 8}{x + 8}

$\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{x - 8}{x + 8}$

Step 2

To write

- \frac{x - 8}{x + 8}

$- \frac{x - 8}{x + 8}$ as a fraction with a common denominator, multiply by

\frac{x + 7}{x + 7}

$\frac{x + 7}{x + 7}$ .

\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{x - 8}{x + 8} \cdot \frac{x + 7}{x + 7}

$\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{x - 8}{x + 8} \cdot \frac{x + 7}{x + 7}$

Step 3

Write each expression with a common denominator of

(x + 7) (x + 8)

$(x + 7) (x + 8)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{x - 8}{x + 8}

$\frac{x - 8}{x + 8}$ by

\frac{x + 7}{x + 7}

$\frac{x + 7}{x + 7}$ .

\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{(x - 8) (x + 7)}{(x + 8) (x + 7)}

$\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{(x - 8) (x + 7)}{(x + 8) (x + 7)}$

Step 3.2

Reorder the factors of

(x + 8) (x + 7)

$(x + 8) (x + 7)$ .

\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{(x - 8) (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{(x - 8) (x + 7)}{(x + 7) (x + 8)}$

\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{(x - 8) (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48}{(x + 7) (x + 8)} - \frac{(x - 8) (x + 7)}{(x + 7) (x + 8)}$

Step 4

Combine the numerators over the common denominator.

\frac{x^{2} - 48 - (x - 8) (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - (x - 8) (x + 7)}{(x + 7) (x + 8)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{x^{2} - 48 + (- x - - 8) (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 + (- x - - 8) (x + 7)}{(x + 7) (x + 8)}$

Step 5.2

Multiply

- 1

$- 1$ by

- 8

$- 8$ .

\frac{x^{2} - 48 + (- x + 8) (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 + (- x + 8) (x + 7)}{(x + 7) (x + 8)}$

Step 5.3

Expand

(- x + 8) (x + 7)

$(- x + 8) (x + 7)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

\frac{x^{2} - 48 - x (x + 7) + 8 (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x (x + 7) + 8 (x + 7)}{(x + 7) (x + 8)}$

Step 5.3.2

Apply the distributive property.

\frac{x^{2} - 48 - x \cdot x - x \cdot 7 + 8 (x + 7)}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x \cdot x - x \cdot 7 + 8 (x + 7)}{(x + 7) (x + 8)}$

Step 5.3.3

Apply the distributive property.

\frac{x^{2} - 48 - x \cdot x - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x \cdot x - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}$

\frac{x^{2} - 48 - x \cdot x - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x \cdot x - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}$

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

$x$ by

x

$x$ by adding the exponents.

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Step 5.4.1.1.1

Move

x

$x$ .

\frac{x^{2} - 48 - (x \cdot x) - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - (x \cdot x) - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}$

Step 5.4.1.1.2

Multiply

x

$x$ by

x

$x$ .

\frac{x^{2} - 48 - x^{2} - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}$

\frac{x^{2} - 48 - x^{2} - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} - x \cdot 7 + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}$

Step 5.4.1.2

Multiply

7

$7$ by

- 1

$- 1$ .

\frac{x^{2} - 48 - x^{2} - 7 x + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} - 7 x + 8 x + 8 \cdot 7}{(x + 7) (x + 8)}$

Step 5.4.1.3

Multiply

8

$8$ by

7

$7$ .

\frac{x^{2} - 48 - x^{2} - 7 x + 8 x + 56}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} - 7 x + 8 x + 56}{(x + 7) (x + 8)}$

\frac{x^{2} - 48 - x^{2} - 7 x + 8 x + 56}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} - 7 x + 8 x + 56}{(x + 7) (x + 8)}$

Step 5.4.2

Add

- 7 x

$- 7 x$ and

8 x

$8 x$ .

\frac{x^{2} - 48 - x^{2} + x + 56}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} + x + 56}{(x + 7) (x + 8)}$

\frac{x^{2} - 48 - x^{2} + x + 56}{(x + 7) (x + 8)}

$\frac{x^{2} - 48 - x^{2} + x + 56}{(x + 7) (x + 8)}$

Step 5.5

Subtract

x^{2}

$x^{2}$ from

x^{2}

$x^{2}$ .

\frac{0 - 48 + x + 56}{(x + 7) (x + 8)}

$\frac{0 - 48 + x + 56}{(x + 7) (x + 8)}$

Step 5.6

Subtract

48

$48$ from

0

$0$ .

\frac{- 48 + x + 56}{(x + 7) (x + 8)}

$\frac{- 48 + x + 56}{(x + 7) (x + 8)}$

Step 5.7

Add

- 48

$- 48$ and

56

$56$ .

\frac{x + 8}{(x + 7) (x + 8)}

$\frac{x + 8}{(x + 7) (x + 8)}$

\frac{x + 8}{(x + 7) (x + 8)}

$\frac{x + 8}{(x + 7) (x + 8)}$

Step 6

Cancel the common factor of

x + 8

$x + 8$ .

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Step 6.1

Cancel the common factor.

\frac{x + 8}{(x + 7) (x + 8)}

$\frac{x + 8}{(x + 7) (x + 8)}$

Step 6.2

Rewrite the expression.

\frac{1}{x + 7}

$\frac{1}{x + 7}$

\frac{1}{x + 7}

$\frac{1}{x + 7}$