Algebra Examples

$\frac{3 x^{2} - 30 x}{4 x} \cdot \frac{x - 10}{x^{2} - 20 x + 100} \div \frac{2 x}{3 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 30 x}{4 x} \cdot \frac{x - 10}{x^{2} - 20 x + 100} \frac{3 x}{2 x}$

Step 2

Factor $3 x$ out of $3 x^{2} - 30 x$ .

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Step 2.1

Factor $3 x$ out of $3 x^{2}$ .

$\frac{3 x (x) - 30 x}{4 x} \cdot \frac{x - 10}{x^{2} - 20 x + 100} \frac{3 x}{2 x}$

Step 2.2

Factor $3 x$ out of $- 30 x$ .

$\frac{3 x (x) + 3 x (- 10)}{4 x} \cdot \frac{x - 10}{x^{2} - 20 x + 100} \frac{3 x}{2 x}$

Step 2.3

Factor $3 x$ out of $3 x (x) + 3 x (- 10)$ .

$\frac{3 x (x - 10)}{4 x} \cdot \frac{x - 10}{x^{2} - 20 x + 100} \frac{3 x}{2 x}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $100$ as $10^{2}$ .

$\frac{3 x (x - 10)}{4 x} \cdot \frac{x - 10}{x^{2} - 20 x + 10^{2}} \frac{3 x}{2 x}$

Step 3.2

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

$20 x = 2 \cdot x \cdot 10$

Step 3.3

Rewrite the polynomial.

$\frac{3 x (x - 10)}{4 x} \cdot \frac{x - 10}{x^{2} - 2 \cdot x \cdot 10 + 10^{2}} \frac{3 x}{2 x}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 10$ .

$\frac{3 x (x - 10)}{4 x} \cdot \frac{x - 10}{{(x - 10)}^{2}} \frac{3 x}{2 x}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 10$ .

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Step 4.1.1

Factor $x - 10$ out of $3 x (x - 10)$ .

$\frac{(x - 10) (3 x)}{4 x} \cdot \frac{x - 10}{{(x - 10)}^{2}} \frac{3 x}{2 x}$

Step 4.1.2

Factor $x - 10$ out of ${(x - 10)}^{2}$ .

$\frac{(x - 10) (3 x)}{4 x} \cdot \frac{x - 10}{(x - 10) (x - 10)} \frac{3 x}{2 x}$

Step 4.1.3

Cancel the common factor.

$\frac{(x - 10) (3 x)}{4 x} \cdot \frac{x - 10}{(x - 10) (x - 10)} \frac{3 x}{2 x}$

Step 4.1.4

Rewrite the expression.

$\frac{3 x}{4 x} \cdot \frac{x - 10}{x - 10} \frac{3 x}{2 x}$

Step 4.2

Multiply $\frac{3 x}{4 x}$ by $\frac{x - 10}{x - 10}$ .

$\frac{3 x (x - 10)}{4 x (x - 10)} \cdot \frac{3 x}{2 x}$

Step 4.3

Cancel the common factor of $x$ .

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Step 4.3.1

Factor $x$ out of $3 x (x - 10)$ .

$\frac{x (3 (x - 10))}{4 x (x - 10)} \cdot \frac{3 x}{2 x}$

Step 4.3.2

Factor $x$ out of $2 x$ .

$\frac{x (3 (x - 10))}{4 x (x - 10)} \cdot \frac{3 x}{x \cdot 2}$

Step 4.3.3

Cancel the common factor.

$\frac{x (3 (x - 10))}{4 x (x - 10)} \cdot \frac{3 x}{x \cdot 2}$

Step 4.3.4

Rewrite the expression.

$\frac{3 (x - 10)}{4 x (x - 10)} \cdot \frac{3 x}{2}$

Step 4.4

Cancel the common factor of $x$ .

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Step 4.4.1

Factor $x$ out of $4 x (x - 10)$ .

$\frac{3 (x - 10)}{x (4 (x - 10))} \cdot \frac{3 x}{2}$

Step 4.4.2

Factor $x$ out of $3 x$ .

$\frac{3 (x - 10)}{x (4 (x - 10))} \cdot \frac{x \cdot 3}{2}$

Step 4.4.3

Cancel the common factor.

$\frac{3 (x - 10)}{x (4 (x - 10))} \cdot \frac{x \cdot 3}{2}$

Step 4.4.4

Rewrite the expression.

$\frac{3 (x - 10)}{4 (x - 10)} \cdot \frac{3}{2}$

Step 4.5

Multiply $\frac{3 (x - 10)}{4 (x - 10)}$ by $\frac{3}{2}$ .

$\frac{3 (x - 10) \cdot 3}{4 (x - 10) \cdot 2}$

Step 4.6

Multiply.

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Step 4.6.1

Multiply $3$ by $3$ .

$\frac{9 (x - 10)}{4 (x - 10) \cdot 2}$

Step 4.6.2

Multiply $2$ by $4$ .

$\frac{9 (x - 10)}{8 (x - 10)}$

Step 4.7

Cancel the common factor of $x - 10$ .

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Step 4.7.1

Cancel the common factor.

$\frac{9 (x - 10)}{8 (x - 10)}$

Step 4.7.2

Rewrite the expression.

$\frac{9}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{8}$

Decimal Form:

$1.125$

Mixed Number Form:

$1 \frac{1}{8}$