Linear Algebra Examples

$\frac{2}{x^{2} - 2 x - 48} + \frac{x}{2 x^{2} + 11 x - 6}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 2 x - 48$ using the AC method.

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Step 1.1.1

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 $- 48$ and whose sum is $- 2$ .

$- 8, 6$

Step 1.1.2

Write the factored form using these integers.

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{2 x^{2} + 11 x - 6}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 11$ .

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Step 1.2.1.1

Factor $11$ out of $11 x$ .

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{2 x^{2} + 11 (x) - 6}$

Step 1.2.1.2

Rewrite $11$ as $- 1$ plus $12$

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{2 x^{2} + (- 1 + 12) x - 6}$

Step 1.2.1.3

Apply the distributive property.

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{2 x^{2} - 1 x + 12 x - 6}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{(2 x^{2} - 1 x) + 12 x - 6}$

Step 1.2.2.2

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

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{x (2 x - 1) + 6 (2 x - 1)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{2}{(x - 8) (x + 6)} + \frac{x}{(2 x - 1) (x + 6)}$

Step 2

To write $\frac{2}{(x - 8) (x + 6)}$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{2}{(x - 8) (x + 6)} \cdot \frac{2 x - 1}{2 x - 1} + \frac{x}{(2 x - 1) (x + 6)}$

Step 3

To write $\frac{x}{(2 x - 1) (x + 6)}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{2}{(x - 8) (x + 6)} \cdot \frac{2 x - 1}{2 x - 1} + \frac{x}{(2 x - 1) (x + 6)} \cdot \frac{x - 8}{x - 8}$

Step 4

Write each expression with a common denominator of $(x - 8) (x + 6) (2 x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{(x - 8) (x + 6)}$ by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{2 (2 x - 1)}{(x - 8) (x + 6) (2 x - 1)} + \frac{x}{(2 x - 1) (x + 6)} \cdot \frac{x - 8}{x - 8}$

Step 4.2

Multiply $\frac{x}{(2 x - 1) (x + 6)}$ by $\frac{x - 8}{x - 8}$ .

$\frac{2 (2 x - 1)}{(x - 8) (x + 6) (2 x - 1)} + \frac{x (x - 8)}{(2 x - 1) (x + 6) (x - 8)}$

Step 4.3

Reorder the factors of $(x - 8) (x + 6) (2 x - 1)$ .

$\frac{2 (2 x - 1)}{(2 x - 1) (x + 6) (x - 8)} + \frac{x (x - 8)}{(2 x - 1) (x + 6) (x - 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (2 x - 1) + x (x - 8)}{(2 x - 1) (x + 6) (x - 8)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 (2 x) + 2 \cdot - 1 + x (x - 8)}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.2

Multiply $2$ by $2$ .

$\frac{4 x + 2 \cdot - 1 + x (x - 8)}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.3

Multiply $2$ by $- 1$ .

$\frac{4 x - 2 + x (x - 8)}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.4

Apply the distributive property.

$\frac{4 x - 2 + x \cdot x + x \cdot - 8}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.5

Multiply $x$ by $x$ .

$\frac{4 x - 2 + x^{2} + x \cdot - 8}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.6

Move $- 8$ to the left of $x$ .

$\frac{4 x - 2 + x^{2} - 8 \cdot x}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.7

Subtract $8 x$ from $4 x$ .

$\frac{- 4 x - 2 + x^{2}}{(2 x - 1) (x + 6) (x - 8)}$

Step 6.8

Reorder terms.

$\frac{x^{2} - 4 x - 2}{(2 x - 1) (x + 6) (x - 8)}$