Algebra Examples

$\frac{x^{2} + 3 x - 28}{2 x^{3}} \div \frac{x^{2} - 7 x + 12}{x - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 3 x - 28}{2 x^{3}} \cdot \frac{x - 3}{x^{2} - 7 x + 12}$

Step 2

Factor $x^{2} + 3 x - 28$ using the AC method.

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Step 2.1

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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 7)}{2 x^{3}} \cdot \frac{x - 3}{x^{2} - 7 x + 12}$

Step 3

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 7)}{2 x^{3}} \cdot \frac{x - 3}{(x - 4) (x - 3)}$

Step 4

Cancel the common factor of $x - 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(x - 4) (x + 7)}{2 x^{3}} \cdot \frac{x - 3}{(x - 4) (x - 3)}$

Step 4.2

Rewrite the expression.

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

Step 5

Multiply $\frac{x + 7}{2 x^{3}}$ by $\frac{x - 3}{x - 3}$ .

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

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{x + 7}{2 x^{3}}$

Step 7

Split the fraction $\frac{x + 7}{2 x^{3}}$ into two fractions.

$\frac{x}{2 x^{3}} + \frac{7}{2 x^{3}}$

Step 8

Cancel the common factor of $x$ and $x^{3}$ .

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Step 8.1

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

$\frac{x^{1}}{2 x^{3}} + \frac{7}{2 x^{3}}$

Step 8.2

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

$\frac{x \cdot 1}{2 x^{3}} + \frac{7}{2 x^{3}}$

Step 8.3

Cancel the common factors.

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Step 8.3.1

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

$\frac{x \cdot 1}{x (2 x^{2})} + \frac{7}{2 x^{3}}$

Step 8.3.2

Cancel the common factor.

$\frac{x \cdot 1}{x (2 x^{2})} + \frac{7}{2 x^{3}}$

Step 8.3.3

Rewrite the expression.

$\frac{1}{2 x^{2}} + \frac{7}{2 x^{3}}$