Algebra Examples

$\frac{3 x^{2}}{3 x} \div \frac{x^{2} - 9}{x^{2} - 10 x + 21}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2}}{3 x} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $3$ .

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Step 2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3 x} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.1.2

Rewrite the expression.

$\frac{x^{2}}{x} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.2.1

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

$\frac{x \cdot x}{x} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Raise $x$ to the power of $1$ .

$\frac{x \cdot x}{x^{1}} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x}{x \cdot 1} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.2.2.3

Cancel the common factor.

$\frac{x \cdot x}{x \cdot 1} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.2.2.4

Rewrite the expression.

$\frac{x}{1} \cdot \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 2.2.2.5

Divide $x$ by $1$ .

$x \frac{x^{2} - 10 x + 21}{x^{2} - 9}$

Step 3

Factor $x^{2} - 10 x + 21$ using the AC method.

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Step 3.1

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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 3.2

Write the factored form using these integers.

$x \frac{(x - 7) (x - 3)}{x^{2} - 9}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$x \frac{(x - 7) (x - 3)}{x^{2} - 3^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 3$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

$x \frac{x - 7}{x + 3}$

Step 5.2

Combine $x$ and $\frac{x - 7}{x + 3}$ .

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