Basic Math Examples

$\frac{4 w}{w^{2} + 2 w - 3} + \frac{2}{1 - w}$

Step 1

Factor $w^{2} + 2 w - 3$ using the AC method.

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Step 1.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.2

Write the factored form using these integers.

$\frac{4 w}{(w - 1) (w + 3)} + \frac{2}{1 - w}$

Step 2

Simplify with factoring out.

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Step 2.1

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

$\frac{4 w}{(w - 1) (w + 3)} + \frac{2}{- 1 (- 1) - w}$

Step 2.2

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

$\frac{4 w}{(w - 1) (w + 3)} + \frac{2}{- 1 (- 1) - (w)}$

Step 2.3

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

$\frac{4 w}{(w - 1) (w + 3)} + \frac{2}{- 1 (- 1 + w)}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{2}{- 1 (- 1 + w)}$ to the numerator.

$\frac{4 w}{(w - 1) (w + 3)} + \frac{- 1 \cdot 2}{- 1 + w}$

Step 2.4.2

Reorder terms.

$\frac{4 w}{(w - 1) (w + 3)} + \frac{- 1 \cdot 2}{w - 1}$

Step 3

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

$\frac{4 w}{(w - 1) (w + 3)} + \frac{- 1 \cdot 2}{w - 1} \cdot \frac{w + 3}{w + 3}$

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{- 1 \cdot 2}{w - 1}$ by $\frac{w + 3}{w + 3}$ .

$\frac{4 w}{(w - 1) (w + 3)} + \frac{- 1 \cdot 2 (w + 3)}{(w - 1) (w + 3)}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{4 w - 1 \cdot 2 (w + 3)}{(w - 1) (w + 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

$\frac{4 w - 2 (w + 3)}{(w - 1) (w + 3)}$

Step 5.2

Apply the distributive property.

$\frac{4 w - 2 w - 2 \cdot 3}{(w - 1) (w + 3)}$

Step 5.3

Multiply $- 2$ by $3$ .

$\frac{4 w - 2 w - 6}{(w - 1) (w + 3)}$

Step 5.4

Subtract $2 w$ from $4 w$ .

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

Step 5.5

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

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Step 5.5.1

Factor $2$ out of $2 w$ .

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

Step 5.5.2

Factor $2$ out of $- 6$ .

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

Step 5.5.3

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

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