Pre-Algebra Examples

$\frac{46200 + v}{22500 - v^{2}} - \frac{46200 - v}{22500 - v^{2}} - \frac{0.5 \cdot (22500 - v^{2})}{22500 - v^{2}}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{46200 + v - (46200 - v) - 0.5 \cdot (22500 - v^{2})}{- v^{2} + 22500}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{46200 + v - 1 \cdot 46200 - - v - 0.5 \cdot (22500 - v^{2})}{- v^{2} + 22500}$

Step 1.2.2

Multiply $- 1$ by $46200$ .

$\frac{46200 + v - 46200 - - v - 0.5 \cdot (22500 - v^{2})}{- v^{2} + 22500}$

Step 1.2.3

Multiply $- - v$ .

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Step 1.2.3.1

Multiply $- 1$ by $- 1$ .

$\frac{46200 + v - 46200 + 1 v - 0.5 \cdot (22500 - v^{2})}{- v^{2} + 22500}$

Step 1.2.3.2

Multiply $v$ by $1$ .

$\frac{46200 + v - 46200 + v - 0.5 \cdot (22500 - v^{2})}{- v^{2} + 22500}$

Step 1.2.4

Apply the distributive property.

$\frac{46200 + v - 46200 + v - 0.5 \cdot 22500 - 0.5 (- v^{2})}{- v^{2} + 22500}$

Step 1.2.5

Multiply $- 0.5$ by $22500$ .

$\frac{46200 + v - 46200 + v - 11250 - 0.5 (- v^{2})}{- v^{2} + 22500}$

Step 1.2.6

Multiply $- 1$ by $- 0.5$ .

$\frac{46200 + v - 46200 + v - 11250 + 0.5 v^{2}}{- v^{2} + 22500}$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $46200 + v - 46200 + v - 11250 + 0.5 v^{2}$ .

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Step 1.3.1.1

Subtract $46200$ from $46200$ .

$\frac{v + 0 + v - 11250 + 0.5 v^{2}}{- v^{2} + 22500}$

Step 1.3.1.2

Add $v$ and $0$ .

$\frac{v + v - 11250 + 0.5 v^{2}}{- v^{2} + 22500}$

Step 1.3.2

Add $v$ and $v$ .

$\frac{2 v - 11250 + 0.5 v^{2}}{- v^{2} + 22500}$

Step 1.3.3

Simplify the expression.

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Step 1.3.3.1

Move $- 11250$ .

$\frac{2 v + 0.5 v^{2} - 11250}{- v^{2} + 22500}$

Step 1.3.3.2

Reorder $2 v$ and $0.5 v^{2}$ .

$\frac{0.5 v^{2} + 2 v - 11250}{- v^{2} + 22500}$

Step 1.3.4

Factor $0.5$ out of $0.5 v^{2} + 2 v - 11250$ .

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Step 1.3.4.1

Factor $0.5$ out of $0.5 v^{2}$ .

$\frac{0.5 (v^{2}) + 2 v - 11250}{- v^{2} + 22500}$

Step 1.3.4.2

Factor $0.5$ out of $2 v$ .

$\frac{0.5 (v^{2}) + 0.5 (4 v) - 11250}{- v^{2} + 22500}$

Step 1.3.4.3

Factor $0.5$ out of $- 11250$ .

$\frac{0.5 v^{2} + 0.5 (4 v) + 0.5 \cdot - 22500}{- v^{2} + 22500}$

Step 1.3.4.4

Factor $0.5$ out of $0.5 v^{2} + 0.5 (4 v)$ .

$\frac{0.5 (v^{2} + 4 v) + 0.5 \cdot - 22500}{- v^{2} + 22500}$

Step 1.3.4.5

Factor $0.5$ out of $0.5 (v^{2} + 4 v) + 0.5 \cdot - 22500$ .

$\frac{0.5 (v^{2} + 4 v - 22500)}{- v^{2} + 22500}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $22500$ as $150^{2}$ .

$\frac{0.5 (v^{2} + 4 v - 22500)}{- v^{2} + 150^{2}}$

Step 2.2

Reorder $- v^{2}$ and $150^{2}$ .

$\frac{0.5 (v^{2} + 4 v - 22500)}{150^{2} - v^{2}}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 150$ and $b = v$ .

$\frac{0.5 (v^{2} + 4 v - 22500)}{(150 + v) (150 - v)}$