Basic Math Examples

$\frac{w - 3}{w^{2} - 36} + \frac{3 - w}{36 - w^{2}}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the denominator.

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Step 1.1.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{w - 3}{w^{2} - 6^{2}} + \frac{3 - w}{36 - w^{2}}$

Step 1.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = w$ and $b = 6$ .

$\frac{w - 3}{(w + 6) (w - 6)} + \frac{3 - w}{36 - w^{2}}$

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

Rewrite $36$ as $6^{2}$ .

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

Step 1.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = w$ .

$\frac{w - 3}{(w + 6) (w - 6)} + \frac{3 - w}{(6 + w) (6 - w)}$

Step 1.2

Simplify terms.

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Step 1.2.1

Reorder terms.

$\frac{w - 3}{(w + 6) (w - 6)} + \frac{3 - w}{(w + 6) (6 - w)}$

Step 1.2.2

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 1.2.3

Factor $- 1$ out of $- w$ .

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

Step 1.2.4

Factor $- 1$ out of $- 1 (- 6) - (w)$ .

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

Step 1.2.5

Simplify the expression.

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Step 1.2.5.1

Move a negative from the denominator of $\frac{3 - w}{(w + 6) (- 1 (- 6 + w))}$ to the numerator.

$\frac{w - 3}{(w + 6) (w - 6)} + \frac{- (3 - w)}{(w + 6) (- 6 + w)}$

Step 1.2.5.2

Reorder terms.

$\frac{w - 3}{(w + 6) (w - 6)} + \frac{- (3 - w)}{(w + 6) (w - 6)}$

Step 1.2.6

Combine the numerators over the common denominator.

$\frac{w - 3 - (3 - w)}{(w + 6) (w - 6)}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $- 1$ by $3$ .

$\frac{w - 3 - 3 - - w}{(w + 6) (w - 6)}$

Step 2.3

Multiply $- - w$ .

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2

Multiply $w$ by $1$ .

$\frac{w - 3 - 3 + w}{(w + 6) (w - 6)}$

Step 2.4

Add $w$ and $w$ .

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

Step 2.5

Subtract $3$ from $- 3$ .

$\frac{2 w - 6}{(w + 6) (w - 6)}$

Step 2.6

Factor $2$ out of $2 w - 6$ .

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Step 2.6.1

Factor $2$ out of $2 w$ .

$\frac{2 (w) - 6}{(w + 6) (w - 6)}$

Step 2.6.2

Factor $2$ out of $- 6$ .

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

Step 2.6.3

Factor $2$ out of $2 w + 2 \cdot - 3$ .

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