Algebra Examples

$\frac{x^{2}}{x - 5} \cdot \frac{x^{2} - 7 x + 10}{x^{2} + 3 x}$

Step 1

Factor $x^{2} - 7 x + 10$ using the AC method.

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Step 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $3 x$ .

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

Step 2.1.3

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

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

Cancel the common factor of $x - 5$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 2.4

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

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