Finite Math Examples

$\frac{y^{2} + 6}{y + 5} - \frac{y - 4}{y - 2}$

Step 1

To write $\frac{y^{2} + 6}{y + 5}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{y^{2} + 6}{y + 5} \cdot \frac{y - 2}{y - 2} - \frac{y - 4}{y - 2}$

Step 2

To write $- \frac{y - 4}{y - 2}$ as a fraction with a common denominator, multiply by $\frac{y + 5}{y + 5}$ .

$\frac{y^{2} + 6}{y + 5} \cdot \frac{y - 2}{y - 2} - \frac{y - 4}{y - 2} \cdot \frac{y + 5}{y + 5}$

Step 3

Write each expression with a common denominator of $(y + 5) (y - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{y^{2} + 6}{y + 5}$ by $\frac{y - 2}{y - 2}$ .

$\frac{(y^{2} + 6) (y - 2)}{(y + 5) (y - 2)} - \frac{y - 4}{y - 2} \cdot \frac{y + 5}{y + 5}$

Step 3.2

Multiply $\frac{y - 4}{y - 2}$ by $\frac{y + 5}{y + 5}$ .

$\frac{(y^{2} + 6) (y - 2)}{(y + 5) (y - 2)} - \frac{(y - 4) (y + 5)}{(y - 2) (y + 5)}$

Step 3.3

Reorder the factors of $(y - 2) (y + 5)$ .

$\frac{(y^{2} + 6) (y - 2)}{(y + 5) (y - 2)} - \frac{(y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(y^{2} + 6) (y - 2) - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(y^{2} + 6) (y - 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{y^{2} (y - 2) + 6 (y - 2) - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.1.2

Apply the distributive property.

$\frac{y^{2} y + y^{2} \cdot - 2 + 6 (y - 2) - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.1.3

Apply the distributive property.

$\frac{y^{2} y + y^{2} \cdot - 2 + 6 y + 6 \cdot - 2 - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 5.2.1.1

Multiply $y^{2}$ by $y$ .

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Step 5.2.1.1.1

Raise $y$ to the power of $1$ .

$\frac{y^{2} y^{1} + y^{2} \cdot - 2 + 6 y + 6 \cdot - 2 - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.2.1.1.2

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

$\frac{y^{2 + 1} + y^{2} \cdot - 2 + 6 y + 6 \cdot - 2 - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.2.1.2

Add $2$ and $1$ .

$\frac{y^{3} + y^{2} \cdot - 2 + 6 y + 6 \cdot - 2 - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.2.2

Move $- 2$ to the left of $y^{2}$ .

$\frac{y^{3} - 2 \cdot y^{2} + 6 y + 6 \cdot - 2 - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.2.3

Multiply $6$ by $- 2$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - (y - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.3

Apply the distributive property.

$\frac{y^{3} - 2 y^{2} + 6 y - 12 + (- y - - 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.4

Multiply $- 1$ by $- 4$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 + (- y + 4) (y + 5)}{(y + 5) (y - 2)}$

Step 5.5

Expand $(- y + 4) (y + 5)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y (y + 5) + 4 (y + 5)}{(y + 5) (y - 2)}$

Step 5.5.2

Apply the distributive property.

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y \cdot y - y \cdot 5 + 4 (y + 5)}{(y + 5) (y - 2)}$

Step 5.5.3

Apply the distributive property.

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y \cdot y - y \cdot 5 + 4 y + 4 \cdot 5}{(y + 5) (y - 2)}$

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $y$ by $y$ by adding the exponents.

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Step 5.6.1.1.1

Move $y$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - (y \cdot y) - y \cdot 5 + 4 y + 4 \cdot 5}{(y + 5) (y - 2)}$

Step 5.6.1.1.2

Multiply $y$ by $y$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y^{2} - y \cdot 5 + 4 y + 4 \cdot 5}{(y + 5) (y - 2)}$

Step 5.6.1.2

Multiply $5$ by $- 1$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y^{2} - 5 y + 4 y + 4 \cdot 5}{(y + 5) (y - 2)}$

Step 5.6.1.3

Multiply $4$ by $5$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y^{2} - 5 y + 4 y + 20}{(y + 5) (y - 2)}$

Step 5.6.2

Add $- 5 y$ and $4 y$ .

$\frac{y^{3} - 2 y^{2} + 6 y - 12 - y^{2} - y + 20}{(y + 5) (y - 2)}$

Step 5.7

Subtract $y^{2}$ from $- 2 y^{2}$ .

$\frac{y^{3} - 3 y^{2} + 6 y - 12 - y + 20}{(y + 5) (y - 2)}$

Step 5.8

Subtract $y$ from $6 y$ .

$\frac{y^{3} - 3 y^{2} + 5 y - 12 + 20}{(y + 5) (y - 2)}$

Step 5.9

Add $- 12$ and $20$ .

$\frac{y^{3} - 3 y^{2} + 5 y + 8}{(y + 5) (y - 2)}$