Algebra Examples

$\frac{x - 25}{5 x - 25} + \frac{3 x + 5}{x^{2} - 5 x}$

Step 1

Simplify each term.

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Step 1.1

Factor $5$ out of $5 x - 25$ .

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Step 1.1.1

Factor $5$ out of $5 x$ .

$\frac{x - 25}{5 (x) - 25} + \frac{3 x + 5}{x^{2} - 5 x}$

Step 1.1.2

Factor $5$ out of $- 25$ .

$\frac{x - 25}{5 x + 5 \cdot - 5} + \frac{3 x + 5}{x^{2} - 5 x}$

Step 1.1.3

Factor $5$ out of $5 x + 5 \cdot - 5$ .

$\frac{x - 25}{5 (x - 5)} + \frac{3 x + 5}{x^{2} - 5 x}$

Step 1.2

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

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Step 1.2.1

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

$\frac{x - 25}{5 (x - 5)} + \frac{3 x + 5}{x \cdot x - 5 x}$

Step 1.2.2

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

$\frac{x - 25}{5 (x - 5)} + \frac{3 x + 5}{x \cdot x + x \cdot - 5}$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{x - 25}{5 (x - 5)} + \frac{3 x + 5}{x (x - 5)}$

Step 2

To write $\frac{x - 25}{5 (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x - 25}{5 (x - 5)} \cdot \frac{x}{x} + \frac{3 x + 5}{x (x - 5)}$

Step 3

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

$\frac{x - 25}{5 (x - 5)} \cdot \frac{x}{x} + \frac{3 x + 5}{x (x - 5)} \cdot \frac{5}{5}$

Step 4

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

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Step 4.1

Multiply $\frac{x - 25}{5 (x - 5)}$ by $\frac{x}{x}$ .

$\frac{(x - 25) x}{5 (x - 5) x} + \frac{3 x + 5}{x (x - 5)} \cdot \frac{5}{5}$

Step 4.2

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

$\frac{(x - 25) x}{5 (x - 5) x} + \frac{(3 x + 5) \cdot 5}{x (x - 5) \cdot 5}$

Step 4.3

Reorder the factors of $5 (x - 5) x$ .

$\frac{(x - 25) x}{5 x (x - 5)} + \frac{(3 x + 5) \cdot 5}{x (x - 5) \cdot 5}$

Step 4.4

Reorder the factors of $x (x - 5) \cdot 5$ .

$\frac{(x - 25) x}{5 x (x - 5)} + \frac{(3 x + 5) \cdot 5}{5 x (x - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 25) x + (3 x + 5) \cdot 5}{5 x (x - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x - 25 x + (3 x + 5) \cdot 5}{5 x (x - 5)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 25 x + (3 x + 5) \cdot 5}{5 x (x - 5)}$

Step 6.3

Apply the distributive property.

$\frac{x^{2} - 25 x + 3 x \cdot 5 + 5 \cdot 5}{5 x (x - 5)}$

Step 6.4

Multiply $5$ by $3$ .

$\frac{x^{2} - 25 x + 15 x + 5 \cdot 5}{5 x (x - 5)}$

Step 6.5

Multiply $5$ by $5$ .

$\frac{x^{2} - 25 x + 15 x + 25}{5 x (x - 5)}$

Step 6.6

Add $- 25 x$ and $15 x$ .

$\frac{x^{2} - 10 x + 25}{5 x (x - 5)}$

Step 6.7

Factor using the perfect square rule.

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Step 6.7.1

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} - 10 x + 5^{2}}{5 x (x - 5)}$

Step 6.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 6.7.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 5 + 5^{2}}{5 x (x - 5)}$

Step 6.7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{{(x - 5)}^{2}}{5 x (x - 5)}$

Step 7

Cancel the common factor of ${(x - 5)}^{2}$ and $x - 5$ .

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Step 7.1

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

$\frac{(x - 5) (x - 5)}{5 x (x - 5)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

$\frac{(x - 5) (x - 5)}{(x - 5) (5 x)}$

Step 7.2.2

Cancel the common factor.

$\frac{(x - 5) (x - 5)}{(x - 5) (5 x)}$

Step 7.2.3

Rewrite the expression.

$\frac{x - 5}{5 x}$