Finite Math Examples

$4 - 8 \frac{x}{x - 5 + 1.5} + {(\frac{x}{x - 5 + 1.5})}^{2}$

Step 1

Add $- 5$ and $1.5$ .

$4 - 8 \frac{x}{x - 3.5} + {(\frac{x}{x - 5 + 1.5})}^{2}$

Step 2

Combine $- 8$ and $\frac{x}{x - 3.5}$ .

$4 + \frac{- 8 x}{x - 3.5} + {(\frac{x}{x - 5 + 1.5})}^{2}$

Step 3

Move the negative in front of the fraction.

$4 - \frac{8 x}{x - 3.5} + {(\frac{x}{x - 5 + 1.5})}^{2}$

Step 4

Add $- 5$ and $1.5$ .

$4 - \frac{8 x}{x - 3.5} + {(\frac{x}{x - 3.5})}^{2}$

Step 5

Apply the product rule to $\frac{x}{x - 3.5}$ .

$4 - \frac{8 x}{x - 3.5} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 6

To write $4$ as a fraction with a common denominator, multiply by $\frac{x - 3.5}{x - 3.5}$ .

$\frac{4 (x - 3.5)}{x - 3.5} - \frac{8 x}{x - 3.5} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 7

Combine the numerators over the common denominator.

$\frac{4 (x - 3.5) - 8 x}{x - 3.5} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 8

To write $\frac{4 (x - 3.5) - 8 x}{x - 3.5}$ as a fraction with a common denominator, multiply by $\frac{x - 3.5}{x - 3.5}$ .

$\frac{4 (x - 3.5) - 8 x}{x - 3.5} \cdot \frac{x - 3.5}{x - 3.5} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 9

Write each expression with a common denominator of ${(x - 3.5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{4 (x - 3.5) - 8 x}{x - 3.5}$ by $\frac{x - 3.5}{x - 3.5}$ .

$\frac{(4 (x - 3.5) - 8 x) (x - 3.5)}{(x - 3.5) (x - 3.5)} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 9.2

Raise $x - 3.5$ to the power of $1$ .

$\frac{(4 (x - 3.5) - 8 x) (x - 3.5)}{{(x - 3.5)}^{1} (x - 3.5)} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 9.3

Raise $x - 3.5$ to the power of $1$ .

$\frac{(4 (x - 3.5) - 8 x) (x - 3.5)}{{(x - 3.5)}^{1} {(x - 3.5)}^{1}} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(4 (x - 3.5) - 8 x) (x - 3.5)}{{(x - 3.5)}^{1 + 1}} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 9.5

Add $1$ and $1$ .

$\frac{(4 (x - 3.5) - 8 x) (x - 3.5)}{{(x - 3.5)}^{2}} + \frac{x^{2}}{{(x - 3.5)}^{2}}$

Step 10

Combine the numerators over the common denominator.

$\frac{(4 (x - 3.5) - 8 x) (x - 3.5) + x^{2}}{{(x - 3.5)}^{2}}$

Step 11

Reorder terms.

$\frac{x^{2} + (4 (x - 3.5) - 8 x) (x - 3.5)}{{(x - 3.5)}^{2}}$

Step 12

Rewrite $\frac{x^{2} + (4 (x - 3.5) - 8 x) (x - 3.5)}{{(x - 3.5)}^{2}}$ in a factored form.

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Step 12.1

Simplify each term.

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Step 12.1.1

Apply the distributive property.

$\frac{x^{2} + (4 x + 4 \cdot - 3.5 - 8 x) (x - 3.5)}{{(x - 3.5)}^{2}}$

Step 12.1.2

Multiply $4$ by $- 3.5$ .

$\frac{x^{2} + (4 x - 14 - 8 x) (x - 3.5)}{{(x - 3.5)}^{2}}$

Step 12.2

Subtract $8 x$ from $4 x$ .

$\frac{x^{2} + (- 4 x - 14) (x - 3.5)}{{(x - 3.5)}^{2}}$

Step 12.3

Expand $(- 4 x - 14) (x - 3.5)$ using the FOIL Method.

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Step 12.3.1

Apply the distributive property.

$\frac{x^{2} - 4 x (x - 3.5) - 14 (x - 3.5)}{{(x - 3.5)}^{2}}$

Step 12.3.2

Apply the distributive property.

$\frac{x^{2} - 4 x \cdot x - 4 x \cdot - 3.5 - 14 (x - 3.5)}{{(x - 3.5)}^{2}}$

Step 12.3.3

Apply the distributive property.

$\frac{x^{2} - 4 x \cdot x - 4 x \cdot - 3.5 - 14 x - 14 \cdot - 3.5}{{(x - 3.5)}^{2}}$

Step 12.4

Simplify and combine like terms.

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Step 12.4.1

Simplify each term.

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Step 12.4.1.1

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

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Step 12.4.1.1.1

Move $x$ .

$\frac{x^{2} - 4 (x \cdot x) - 4 x \cdot - 3.5 - 14 x - 14 \cdot - 3.5}{{(x - 3.5)}^{2}}$

Step 12.4.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 4 x^{2} - 4 x \cdot - 3.5 - 14 x - 14 \cdot - 3.5}{{(x - 3.5)}^{2}}$

Step 12.4.1.2

Multiply $- 3.5$ by $- 4$ .

$\frac{x^{2} - 4 x^{2} + 14 x - 14 x - 14 \cdot - 3.5}{{(x - 3.5)}^{2}}$

Step 12.4.1.3

Multiply $- 14$ by $- 3.5$ .

$\frac{x^{2} - 4 x^{2} + 14 x - 14 x + 49}{{(x - 3.5)}^{2}}$

Step 12.4.2

Subtract $14 x$ from $14 x$ .

$\frac{x^{2} - 4 x^{2} + 0 x + 49}{{(x - 3.5)}^{2}}$

Step 12.5

Multiply $0$ by $x$ .

$\frac{x^{2} - 4 x^{2} + 0 + 49}{{(x - 3.5)}^{2}}$

Step 12.6

Add $- 4 x^{2}$ and $0$ .

$\frac{x^{2} - 4 x^{2} + 49}{{(x - 3.5)}^{2}}$

Step 12.7

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

$\frac{- 3 x^{2} + 49}{{(x - 3.5)}^{2}}$

Step 12.8

Factor $1$ out of $- 3 x^{2} + 49$ .

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Step 12.8.1

Factor $1$ out of $- 3 x^{2}$ .

$\frac{1 (- 3 x^{2}) + 49}{{(x - 3.5)}^{2}}$

Step 12.8.2

Rewrite $49$ as $1 (49)$ .

$\frac{1 (- 3 x^{2}) + 1 (49)}{{(x - 3.5)}^{2}}$

Step 12.8.3

Factor $1$ out of $1 (- 3 x^{2}) + 1 (49)$ .

$\frac{1 (- 3 x^{2} + 49)}{{(x - 3.5)}^{2}}$