Algebra Examples

$q (3) = \frac{1}{x^{2} - 9}$

Step 1

Rewrite the equation as $\frac{1}{x^{2} - 9} = q (3)$ .

$\frac{1}{x^{2} - 9} = q (3)$

Step 2

Factor each term.

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Step 2.1

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

$\frac{1}{x^{2} - 3^{2}} = q (3)$

Step 2.2

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 $b = 3$ .

$\frac{1}{(x + 3) (x - 3)} = q (3)$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x + 3) (x - 3), 1$

Step 3.2

The LCM of one and any expression is the expression.

$(x + 3) (x - 3)$

Step 4

Multiply each term in $\frac{1}{(x + 3) (x - 3)} = q (3)$ by $(x + 3) (x - 3)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{1}{(x + 3) (x - 3)} = q (3)$ by $(x + 3) (x - 3)$ .

$\frac{1}{(x + 3) (x - 3)} ((x + 3) (x - 3)) = q (3) ((x + 3) (x - 3))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $(x + 3) (x - 3)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{1}{(x + 3) (x - 3)} ((x + 3) (x - 3)) = q (3) ((x + 3) (x - 3))$

Step 4.2.1.2

Rewrite the expression.

$1 = q (3) ((x + 3) (x - 3))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move $3$ to the left of $q$ .

$1 = 3 q ((x + 3) (x - 3))$

Step 4.3.2

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

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Step 4.3.2.1

Apply the distributive property.

$1 = 3 q (x (x - 3) + 3 (x - 3))$

Step 4.3.2.2

Apply the distributive property.

$1 = 3 q (x \cdot x + x \cdot - 3 + 3 (x - 3))$

Step 4.3.2.3

Apply the distributive property.

$1 = 3 q (x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3)$

Step 4.3.3

Simplify terms.

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Step 4.3.3.1

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

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Step 4.3.3.1.1

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

$1 = 3 q (x \cdot x - 3 x + 3 x + 3 \cdot - 3)$

Step 4.3.3.1.2

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

$1 = 3 q (x \cdot x + 0 + 3 \cdot - 3)$

Step 4.3.3.1.3

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

$1 = 3 q (x \cdot x + 3 \cdot - 3)$

Step 4.3.3.2

Simplify each term.

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Step 4.3.3.2.1

Multiply $x$ by $x$ .

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

Step 4.3.3.2.2

Multiply $3$ by $- 3$ .

$1 = 3 q (x^{2} - 9)$

Step 4.3.3.3

Simplify by multiplying through.

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Step 4.3.3.3.1

Apply the distributive property.

$1 = 3 q x^{2} + 3 q \cdot - 9$

Step 4.3.3.3.2

Multiply $- 9$ by $3$ .

$1 = 3 q x^{2} - 27 q$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $3 q x^{2} - 27 q = 1$ .

$3 q x^{2} - 27 q = 1$

Step 5.2

Add $27 q$ to both sides of the equation.

$3 q x^{2} = 1 + 27 q$

Step 5.3

Divide each term in $3 q x^{2} = 1 + 27 q$ by $3 q$ and simplify.

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Step 5.3.1

Divide each term in $3 q x^{2} = 1 + 27 q$ by $3 q$ .

$\frac{3 q x^{2}}{3 q} = \frac{1}{3 q} + \frac{27 q}{3 q}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 q x^{2}}{3 q} = \frac{1}{3 q} + \frac{27 q}{3 q}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{q x^{2}}{q} = \frac{1}{3 q} + \frac{27 q}{3 q}$

Step 5.3.2.2

Cancel the common factor of $q$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{q x^{2}}{q} = \frac{1}{3 q} + \frac{27 q}{3 q}$

Step 5.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{3 q} + \frac{27 q}{3 q}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $27$ and $3$ .

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Step 5.3.3.1.1.1

Factor $3$ out of $27 q$ .

$x^{2} = \frac{1}{3 q} + \frac{3 (9 q)}{3 q}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Factor $3$ out of $3 q$ .

$x^{2} = \frac{1}{3 q} + \frac{3 (9 q)}{3 (q)}$

Step 5.3.3.1.1.2.2

Cancel the common factor.

$x^{2} = \frac{1}{3 q} + \frac{3 (9 q)}{3 q}$

Step 5.3.3.1.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{3 q} + \frac{9 q}{q}$

Step 5.3.3.1.2

Cancel the common factor of $q$ .

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Step 5.3.3.1.2.1

Cancel the common factor.

$x^{2} = \frac{1}{3 q} + \frac{9 q}{q}$

Step 5.3.3.1.2.2

Divide $9$ by $1$ .

$x^{2} = \frac{1}{3 q} + 9$

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{3 q} + 9}$

Step 5.5

Simplify $\pm \sqrt{\frac{1}{3 q} + 9}$ .

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Step 5.5.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{3 q}{3 q}$ .

$x = \pm \sqrt{\frac{1}{3 q} + 9 \cdot \frac{3 q}{3 q}}$

Step 5.5.2

Combine $9$ and $\frac{3 q}{3 q}$ .

$x = \pm \sqrt{\frac{1}{3 q} + \frac{9 (3 q)}{3 q}}$

Step 5.5.3

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{1 + 9 (3 q)}{3 q}}$

Step 5.5.4

Multiply $9$ by $3$ .

$x = \pm \sqrt{\frac{1 + 27 q}{3 q}}$

Step 5.5.5

Rewrite $\sqrt{\frac{1 + 27 q}{3 q}}$ as $\frac{\sqrt{1 + 27 q}}{\sqrt{3 q}}$ .

$x = \pm \frac{\sqrt{1 + 27 q}}{\sqrt{3 q}}$

Step 5.5.6

Multiply $\frac{\sqrt{1 + 27 q}}{\sqrt{3 q}}$ by $\frac{\sqrt{3 q}}{\sqrt{3 q}}$ .

$x = \pm \frac{\sqrt{1 + 27 q}}{\sqrt{3 q}} \cdot \frac{\sqrt{3 q}}{\sqrt{3 q}}$

Step 5.5.7

Combine and simplify the denominator.

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Step 5.5.7.1

Multiply $\frac{\sqrt{1 + 27 q}}{\sqrt{3 q}}$ by $\frac{\sqrt{3 q}}{\sqrt{3 q}}$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{\sqrt{3 q} \sqrt{3 q}}$

Step 5.5.7.2

Raise $\sqrt{3 q}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{\sqrt{3 q}}^{1} \sqrt{3 q}}$

Step 5.5.7.3

Raise $\sqrt{3 q}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{\sqrt{3 q}}^{1} {\sqrt{3 q}}^{1}}$

Step 5.5.7.4

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

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{\sqrt{3 q}}^{1 + 1}}$

Step 5.5.7.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{\sqrt{3 q}}^{2}}$

Step 5.5.7.6

Rewrite ${\sqrt{3 q}}^{2}$ as $3 q$ .

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Step 5.5.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 q}$ as ${(3 q)}^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{({(3 q)}^{\frac{1}{2}})}^{2}}$

Step 5.5.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{(3 q)}^{\frac{1}{2} \cdot 2}}$

Step 5.5.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{(3 q)}^{\frac{2}{2}}}$

Step 5.5.7.6.4

Cancel the common factor of $2$ .

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Step 5.5.7.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{(3 q)}^{\frac{2}{2}}}$

Step 5.5.7.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{{(3 q)}^{1}}$

Step 5.5.7.6.5

Simplify.

$x = \pm \frac{\sqrt{1 + 27 q} \sqrt{3 q}}{3 q}$

Step 5.5.8

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 + 27 q) \cdot 3 q}}{3 q}$

Step 5.5.9

Reorder factors in $\pm \frac{\sqrt{(1 + 27 q) \cdot 3 q}}{3 q}$ .

$x = \pm \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

$x = - \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

$x = \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

$x = - \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

$x = \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

$x = - \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$

Step 6

To rewrite as a function of $q$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $q$ is on the other side.

$f (q) = \frac{\sqrt{3 q (1 + 27 q)}}{3 q}, - \frac{\sqrt{3 q (1 + 27 q)}}{3 q}$