Basic Math Examples

$\frac{6 π}{π^{2} - 1} - \frac{3}{π^{2} - π - 2}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{6 π}{π^{2} - 1^{2}} - \frac{3}{π^{2} - π - 2}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = π$ and $b = 1$ .

$\frac{6 π}{(π + 1) (π - 1)} - \frac{3}{π^{2} - π - 2}$

Step 1.2

Factor $π^{2} - π - 2$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{6 π}{(π + 1) (π - 1)} - \frac{3}{(π - 2) (π + 1)}$

Step 2

To write $\frac{6 π}{(π + 1) (π - 1)}$ as a fraction with a common denominator, multiply by $\frac{π - 2}{π - 2}$ .

$\frac{6 π}{(π + 1) (π - 1)} \cdot \frac{π - 2}{π - 2} - \frac{3}{(π - 2) (π + 1)}$

Step 3

To write $- \frac{3}{(π - 2) (π + 1)}$ as a fraction with a common denominator, multiply by $\frac{π - 1}{π - 1}$ .

$\frac{6 π}{(π + 1) (π - 1)} \cdot \frac{π - 2}{π - 2} - \frac{3}{(π - 2) (π + 1)} \cdot \frac{π - 1}{π - 1}$

Step 4

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

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Step 4.1

Multiply $\frac{6 π}{(π + 1) (π - 1)}$ by $\frac{π - 2}{π - 2}$ .

$\frac{6 π (π - 2)}{(π + 1) (π - 1) (π - 2)} - \frac{3}{(π - 2) (π + 1)} \cdot \frac{π - 1}{π - 1}$

Step 4.2

Multiply $\frac{3}{(π - 2) (π + 1)}$ by $\frac{π - 1}{π - 1}$ .

$\frac{6 π (π - 2)}{(π + 1) (π - 1) (π - 2)} - \frac{3 (π - 1)}{(π - 2) (π + 1) (π - 1)}$

Step 4.3

Reorder the factors of $(π - 2) (π + 1) (π - 1)$ .

$\frac{6 π (π - 2)}{(π + 1) (π - 1) (π - 2)} - \frac{3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 π (π - 2) - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{6 π π + 6 π \cdot - 2 - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6.2

Multiply $6 π π$ .

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Step 6.2.1

Raise $π$ to the power of $1$ .

$\frac{6 (π^{1} π) + 6 π \cdot - 2 - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6.2.2

Raise $π$ to the power of $1$ .

$\frac{6 (π^{1} π^{1}) + 6 π \cdot - 2 - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6.2.3

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

$\frac{6 π^{1 + 1} + 6 π \cdot - 2 - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6.2.4

Add $1$ and $1$ .

$\frac{6 π^{2} + 6 π \cdot - 2 - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6.3

Multiply $- 2$ by $6$ .

$\frac{6 π^{2} - 12 π - 3 (π - 1)}{(π + 1) (π - 1) (π - 2)}$

Step 6.4

Apply the distributive property.

$\frac{6 π^{2} - 12 π - 3 π - 3 \cdot - 1}{(π + 1) (π - 1) (π - 2)}$

Step 6.5

Multiply $- 3$ by $- 1$ .

$\frac{6 π^{2} - 12 π - 3 π + 3}{(π + 1) (π - 1) (π - 2)}$

Step 6.6

Subtract $3 π$ from $- 12 π$ .

$\frac{6 π^{2} - 15 π + 3}{(π + 1) (π - 1) (π - 2)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{6 π^{2} - 15 π + 3}{(π + 1) (π - 1) (π - 2)}$

Decimal Form:

$1.49066945 \dots$