Algebra Examples

$\frac{2 x}{x^{2} - 49} - \frac{3}{(x - 4) (x - 7)}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{2 x}{x^{2} - 7^{2}} - \frac{3}{(x - 4) (x - 7)}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{2 x}{(x + 7) (x - 7)} - \frac{3}{(x - 4) (x - 7)}$

Step 2

To write $\frac{2 x}{(x + 7) (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{2 x}{(x + 7) (x - 7)} \cdot \frac{x - 4}{x - 4} - \frac{3}{(x - 4) (x - 7)}$

Step 3

To write $- \frac{3}{(x - 4) (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{2 x}{(x + 7) (x - 7)} \cdot \frac{x - 4}{x - 4} - \frac{3}{(x - 4) (x - 7)} \cdot \frac{x + 7}{x + 7}$

Step 4

Write each expression with a common denominator of $(x + 7) (x - 7) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 x}{(x + 7) (x - 7)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{2 x (x - 4)}{(x + 7) (x - 7) (x - 4)} - \frac{3}{(x - 4) (x - 7)} \cdot \frac{x + 7}{x + 7}$

Step 4.2

Multiply $\frac{3}{(x - 4) (x - 7)}$ by $\frac{x + 7}{x + 7}$ .

$\frac{2 x (x - 4)}{(x + 7) (x - 7) (x - 4)} - \frac{3 (x + 7)}{(x - 4) (x - 7) (x + 7)}$

Step 4.3

Reorder the factors of $(x + 7) (x - 7) (x - 4)$ .

$\frac{2 x (x - 4)}{(x + 7) (x - 4) (x - 7)} - \frac{3 (x + 7)}{(x - 4) (x - 7) (x + 7)}$

Step 4.4

Reorder the factors of $(x - 4) (x - 7) (x + 7)$ .

$\frac{2 x (x - 4)}{(x + 7) (x - 4) (x - 7)} - \frac{3 (x + 7)}{(x + 7) (x - 4) (x - 7)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 x (x - 4) - 3 (x + 7)}{(x + 7) (x - 4) (x - 7)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 4 - 3 (x + 7)}{(x + 7) (x - 4) (x - 7)}$

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 4 - 3 (x + 7)}{(x + 7) (x - 4) (x - 7)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 4 - 3 (x + 7)}{(x + 7) (x - 4) (x - 7)}$

Step 6.3

Multiply $- 4$ by $2$ .

$\frac{2 x^{2} - 8 x - 3 (x + 7)}{(x + 7) (x - 4) (x - 7)}$

Step 6.4

Apply the distributive property.

$\frac{2 x^{2} - 8 x - 3 x - 3 \cdot 7}{(x + 7) (x - 4) (x - 7)}$

Step 6.5

Multiply $- 3$ by $7$ .

$\frac{2 x^{2} - 8 x - 3 x - 21}{(x + 7) (x - 4) (x - 7)}$

Step 6.6

Subtract $3 x$ from $- 8 x$ .

$\frac{2 x^{2} - 11 x - 21}{(x + 7) (x - 4) (x - 7)}$

Step 6.7

Factor by grouping.

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Step 6.7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 21 = - 42$ and whose sum is $b = - 11$ .

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Step 6.7.1.1

Factor $- 11$ out of $- 11 x$ .

$\frac{2 x^{2} - 11 (x) - 21}{(x + 7) (x - 4) (x - 7)}$

Step 6.7.1.2

Rewrite $- 11$ as $3$ plus $- 14$

$\frac{2 x^{2} + (3 - 14) x - 21}{(x + 7) (x - 4) (x - 7)}$

Step 6.7.1.3

Apply the distributive property.

$\frac{2 x^{2} + 3 x - 14 x - 21}{(x + 7) (x - 4) (x - 7)}$

Step 6.7.2

Factor out the greatest common factor from each group.

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Step 6.7.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 3 x) - 14 x - 21}{(x + 7) (x - 4) (x - 7)}$

Step 6.7.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x + 3) - 7 (2 x + 3)}{(x + 7) (x - 4) (x - 7)}$

Step 6.7.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$\frac{(2 x + 3) (x - 7)}{(x + 7) (x - 4) (x - 7)}$

Step 7

Cancel the common factor of $x - 7$ .

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Step 7.1

Cancel the common factor.

$\frac{(2 x + 3) (x - 7)}{(x + 7) (x - 4) (x - 7)}$

Step 7.2

Rewrite the expression.

$\frac{2 x + 3}{(x + 7) (x - 4)}$