Algebra Examples

$\frac{x^{2} + 3 x - 18}{3 x^{2}} \div (x + 6)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{x^{2} + 3 x - 18}{3 x^{2}}}{x + 6}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 3 x - 18}{3 x^{2}} \cdot \frac{1}{x + 6}$

Step 3

Factor $x^{2} + 3 x - 18$ 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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 6)}{3 x^{2}} \cdot \frac{1}{x + 6}$

Step 4

Cancel the common factor of $x + 6$ .

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Step 4.1

Factor $x + 6$ out of $(x - 3) (x + 6)$ .

$\frac{(x + 6) (x - 3)}{3 x^{2}} \cdot \frac{1}{x + 6}$

Step 4.2

Cancel the common factor.

$\frac{(x + 6) (x - 3)}{3 x^{2}} \cdot \frac{1}{x + 6}$

Step 4.3

Rewrite the expression.

$\frac{x - 3}{3 x^{2}}$

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Cancel the common factors.

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Step 6.3.1

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

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

Step 6.3.2

Cancel the common factor.

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

Step 6.3.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $- 3$ and $3$ .

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Step 7.1

Factor $3$ out of $- 3$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Move the negative in front of the fraction.

$\frac{1}{3 x} - \frac{1}{x^{2}}$