Algebra Examples

$\frac{4 x^{2} - 20 x + 25}{x^{2} - 4 x} \cdot \frac{3 x - 12}{2 x - 5}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{{(2 x)}^{2} - 20 x + 25}{x^{2} - 4 x} \cdot \frac{3 x - 12}{2 x - 5}$

Step 1.2

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

$\frac{{(2 x)}^{2} - 20 x + 5^{2}}{x^{2} - 4 x} \cdot \frac{3 x - 12}{2 x - 5}$

Step 1.3

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

$20 x = 2 \cdot (2 x) \cdot 5$

Step 1.4

Rewrite the polynomial.

$\frac{{(2 x)}^{2} - 2 \cdot (2 x) \cdot 5 + 5^{2}}{x^{2} - 4 x} \cdot \frac{3 x - 12}{2 x - 5}$

Step 1.5

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

$\frac{{(2 x - 5)}^{2}}{x^{2} - 4 x} \cdot \frac{3 x - 12}{2 x - 5}$

Step 2

Simplify terms.

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Step 2.1

Factor $x$ out of $x^{2} - 4 x$ .

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Step 2.1.1

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

$\frac{{(2 x - 5)}^{2}}{x \cdot x - 4 x} \cdot \frac{3 x - 12}{2 x - 5}$

Step 2.1.2

Factor $x$ out of $- 4 x$ .

$\frac{{(2 x - 5)}^{2}}{x \cdot x + x \cdot - 4} \cdot \frac{3 x - 12}{2 x - 5}$

Step 2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$\frac{{(2 x - 5)}^{2}}{x (x - 4)} \cdot \frac{3 x - 12}{2 x - 5}$

Step 2.2

Factor $3$ out of $3 x - 12$ .

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Step 2.2.1

Factor $3$ out of $3 x$ .

$\frac{{(2 x - 5)}^{2}}{x (x - 4)} \cdot \frac{3 (x) - 12}{2 x - 5}$

Step 2.2.2

Factor $3$ out of $- 12$ .

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

Step 2.2.3

Factor $3$ out of $3 x + 3 \cdot - 4$ .

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

Step 2.3

Cancel the common factor of $2 x - 5$ .

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor.

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

Step 2.3.3

Rewrite the expression.

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

Step 2.4

Cancel the common factor of $x - 4$ .

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Step 2.4.1

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

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

Step 2.4.2

Factor $x - 4$ out of $3 (x - 4)$ .

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

Step 2.4.3

Cancel the common factor.

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

Step 2.4.4

Rewrite the expression.

$\frac{2 x - 5}{x} \cdot 3$

Step 2.5

Combine $\frac{2 x - 5}{x}$ and $3$ .

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

Step 2.6

Move $3$ to the left of $2 x - 5$ .

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