Algebra Examples

$\frac{4}{x^{2} - 3 x} - \frac{1}{x^{2} - 9} = 0$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{4}{x^{2} - 3 x} - \frac{1}{x^{2} - 3^{2}} = 0$

Step 1.2

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 $b = 3$ .

$\frac{4}{x^{2} - 3 x} - \frac{1}{(x + 3) (x - 3)} = 0$

Step 2

To write $\frac{4}{x^{2} - 3 x}$ as a fraction with a common denominator, multiply by $\frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$ .

$\frac{4}{x^{2} - 3 x} \cdot \frac{(x + 3) (x - 3)}{(x + 3) (x - 3)} - \frac{1}{(x + 3) (x - 3)} = 0$

Step 3

To write $- \frac{1}{(x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 3 x}{x^{2} - 3 x}$ .

$\frac{4}{x^{2} - 3 x} \cdot \frac{(x + 3) (x - 3)}{(x + 3) (x - 3)} - \frac{1}{(x + 3) (x - 3)} \cdot \frac{x^{2} - 3 x}{x^{2} - 3 x} = 0$

Step 4

Write each expression with a common denominator of $(x^{2} - 3 x) (x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4}{x^{2} - 3 x}$ by $\frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$ .

$\frac{4 ((x + 3) (x - 3))}{(x^{2} - 3 x) ((x + 3) (x - 3))} - \frac{1}{(x + 3) (x - 3)} \cdot \frac{x^{2} - 3 x}{x^{2} - 3 x} = 0$

Step 4.2

Multiply $\frac{1}{(x + 3) (x - 3)}$ by $\frac{x^{2} - 3 x}{x^{2} - 3 x}$ .

$\frac{4 ((x + 3) (x - 3))}{(x^{2} - 3 x) ((x + 3) (x - 3))} - \frac{x^{2} - 3 x}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 4.3

Reorder the factors of $(x^{2} - 3 x) ((x + 3) (x - 3))$ .

$\frac{4 ((x + 3) (x - 3))}{(x + 3) (x - 3) (x^{2} - 3 x)} - \frac{x^{2} - 3 x}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{4 ((x + 3) (x - 3)) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6

Rewrite $\frac{4 ((x + 3) (x - 3)) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)}$ in a factored form.

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Step 6.1

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{4 (x (x - 3) + 3 (x - 3)) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.1.2

Apply the distributive property.

$\frac{4 (x \cdot x + x \cdot - 3 + 3 (x - 3)) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.1.3

Apply the distributive property.

$\frac{4 (x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.2

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 6.2.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$\frac{4 (x \cdot x - 3 x + 3 x + 3 \cdot - 3) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.2.2

Add $- 3 x$ and $3 x$ .

$\frac{4 (x \cdot x + 0 + 3 \cdot - 3) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.2.3

Add $x \cdot x$ and $0$ .

$\frac{4 (x \cdot x + 3 \cdot - 3) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.3

Simplify each term.

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Step 6.3.1

Multiply $x$ by $x$ .

$\frac{4 (x^{2} + 3 \cdot - 3) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.3.2

Multiply $3$ by $- 3$ .

$\frac{4 (x^{2} - 9) - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.4

Apply the distributive property.

$\frac{4 x^{2} + 4 \cdot - 9 - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.5

Multiply $4$ by $- 9$ .

$\frac{4 x^{2} - 36 - (x^{2} - 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.6

Apply the distributive property.

$\frac{4 x^{2} - 36 - x^{2} - (- 3 x)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.7

Multiply $- 3$ by $- 1$ .

$\frac{4 x^{2} - 36 - x^{2} + 3 x}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.8

Subtract $x^{2}$ from $4 x^{2}$ .

$\frac{3 x^{2} - 36 + 3 x}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.9

Reorder terms.

$\frac{3 x^{2} + 3 x - 36}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10

Rewrite $3 x^{2} + 3 x - 36$ in a factored form.

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Step 6.10.1

Factor $3$ out of $3 x^{2} + 3 x - 36$ .

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Step 6.10.1.1

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

$\frac{3 (x^{2}) + 3 x - 36}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10.1.2

Factor $3$ out of $3 x$ .

$\frac{3 (x^{2}) + 3 (x) - 36}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10.1.3

Factor $3$ out of $- 36$ .

$\frac{3 (x^{2}) + 3 x + 3 \cdot - 12}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10.1.4

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

$\frac{3 (x^{2} + x) + 3 \cdot - 12}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10.1.5

Factor $3$ out of $3 (x^{2} + x) + 3 \cdot - 12$ .

$\frac{3 (x^{2} + x - 12)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10.2

Factor.

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Step 6.10.2.1

Factor $x^{2} + x - 12$ using the AC method.

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Step 6.10.2.1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 6.10.2.1.2

Write the factored form using these integers.

$\frac{3 ((x - 3) (x + 4))}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.10.2.2

Remove unnecessary parentheses.

$\frac{3 (x - 3) (x + 4)}{(x + 3) (x - 3) (x^{2} - 3 x)} = 0$

Step 6.11

Factor.

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Step 6.11.1

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

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Step 6.11.1.1

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

$\frac{3 (x - 3) (x + 4)}{(x + 3) (x - 3) (x \cdot x - 3 x)} = 0$

Step 6.11.1.2

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

$\frac{3 (x - 3) (x + 4)}{(x + 3) (x - 3) (x \cdot x + x \cdot - 3)} = 0$

Step 6.11.1.3

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

$\frac{3 (x - 3) (x + 4)}{(x + 3) (x - 3) (x (x - 3))} = 0$

Step 6.11.2

Remove unnecessary parentheses.

$\frac{3 (x - 3) (x + 4)}{(x + 3) (x - 3) x (x - 3)} = 0$

Step 6.12

Combine exponents.

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Step 6.12.1

Raise $x - 3$ to the power of $1$ .

$\frac{3 (x - 3) (x + 4)}{(x + 3) x ((x - 3) (x - 3))} = 0$

Step 6.12.2

Raise $x - 3$ to the power of $1$ .

$\frac{3 (x - 3) (x + 4)}{(x + 3) x ((x - 3) (x - 3))} = 0$

Step 6.12.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (x - 3) (x + 4)}{(x + 3) x {(x - 3)}^{1 + 1}} = 0$

Step 6.12.4

Add $1$ and $1$ .

$\frac{3 (x - 3) (x + 4)}{(x + 3) x {(x - 3)}^{2}} = 0$

Step 6.13

Reduce the expression $\frac{3 (x - 3) (x + 4)}{(x + 3) x {(x - 3)}^{2}}$ by cancelling the common factors.

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Step 6.13.1

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

$\frac{(x - 3) (3 (x + 4))}{(x + 3) x {(x - 3)}^{2}} = 0$

Step 6.13.2

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

$\frac{(x - 3) (3 (x + 4))}{(x - 3) ((x + 3) x (x - 3))} = 0$

Step 6.13.3

Cancel the common factor.

$\frac{(x - 3) (3 (x + 4))}{(x - 3) ((x + 3) x (x - 3))} = 0$

Step 6.13.4

Rewrite the expression.

$\frac{3 (x + 4)}{(x + 3) x (x - 3)} = 0$

Step 7

Set the numerator equal to zero.

$3 (x + 4) = 0$

Step 8

Solve the equation for $x$ .

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Step 8.1

Divide each term in $3 (x + 4) = 0$ by $3$ and simplify.

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Step 8.1.1

Divide each term in $3 (x + 4) = 0$ by $3$ .

$\frac{3 (x + 4)}{3} = \frac{0}{3}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Cancel the common factor of $3$ .

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Step 8.1.2.1.1

Cancel the common factor.

$\frac{3 (x + 4)}{3} = \frac{0}{3}$

Step 8.1.2.1.2

Divide $x + 4$ by $1$ .

$x + 4 = \frac{0}{3}$

Step 8.1.3

Simplify the right side.

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Step 8.1.3.1

Divide $0$ by $3$ .

$x + 4 = 0$

Step 8.2

Subtract $4$ from both sides of the equation.

$x = - 4$