Algebra Examples

$\frac{3}{5 x^{2} - 20 x} + \frac{2}{x^{2} + 3 x - 28}$

Step 1

Simplify each term.

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Step 1.1

Factor $5 x$ out of $5 x^{2} - 20 x$ .

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Step 1.1.1

Factor $5 x$ out of $5 x^{2}$ .

$\frac{3}{5 x (x) - 20 x} + \frac{2}{x^{2} + 3 x - 28}$

Step 1.1.2

Factor $5 x$ out of $- 20 x$ .

$\frac{3}{5 x (x) + 5 x (- 4)} + \frac{2}{x^{2} + 3 x - 28}$

Step 1.1.3

Factor $5 x$ out of $5 x (x) + 5 x (- 4)$ .

$\frac{3}{5 x (x - 4)} + \frac{2}{x^{2} + 3 x - 28}$

Step 1.2

Factor $x^{2} + 3 x - 28$ 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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 1.2.2

Write the factored form using these integers.

$\frac{3}{5 x (x - 4)} + \frac{2}{(x - 4) (x + 7)}$

Step 2

To write $\frac{3}{5 x (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{3}{5 x (x - 4)} \cdot \frac{x + 7}{x + 7} + \frac{2}{(x - 4) (x + 7)}$

Step 3

To write $\frac{2}{(x - 4) (x + 7)}$ as a fraction with a common denominator, multiply by $\frac{5 x}{5 x}$ .

$\frac{3}{5 x (x - 4)} \cdot \frac{x + 7}{x + 7} + \frac{2}{(x - 4) (x + 7)} \cdot \frac{5 x}{5 x}$

Step 4

Write each expression with a common denominator of $5 x (x - 4) (x + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3}{5 x (x - 4)}$ by $\frac{x + 7}{x + 7}$ .

$\frac{3 (x + 7)}{5 x (x - 4) (x + 7)} + \frac{2}{(x - 4) (x + 7)} \cdot \frac{5 x}{5 x}$

Step 4.2

Multiply $\frac{2}{(x - 4) (x + 7)}$ by $\frac{5 x}{5 x}$ .

$\frac{3 (x + 7)}{5 x (x - 4) (x + 7)} + \frac{2 (5 x)}{(x - 4) (x + 7) (5 x)}$

Step 4.3

Reorder the factors of $5 x (x - 4) (x + 7)$ .

$\frac{3 (x + 7)}{5 x (x + 7) (x - 4)} + \frac{2 (5 x)}{(x - 4) (x + 7) (5 x)}$

Step 4.4

Reorder the factors of $(x - 4) (x + 7) (5 x)$ .

$\frac{3 (x + 7)}{5 x (x + 7) (x - 4)} + \frac{2 (5 x)}{5 x (x + 7) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 (x + 7) + 2 (5 x)}{5 x (x + 7) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 7 + 2 \cdot 5 x}{5 x (x + 7) (x - 4)}$

Step 6.2

Multiply $3$ by $7$ .

$\frac{3 x + 21 + 2 \cdot 5 x}{5 x (x + 7) (x - 4)}$

Step 6.3

Multiply $2$ by $5$ .

$\frac{3 x + 21 + 10 x}{5 x (x + 7) (x - 4)}$

Step 6.4

Add $3 x$ and $10 x$ .

$\frac{13 x + 21}{5 x (x + 7) (x - 4)}$