Precalculus Examples

$\frac{\frac{3}{2} a^{2} - 2 a b + \frac{2}{3} b^{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1

Set the denominator in $\frac{\frac{3}{2} a^{2} - 2 a b + \frac{2}{3} b^{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}}$ equal to $0$ to find where the expression is undefined.

$\frac{1}{4} a^{2} - \frac{1}{9} b^{2} = 0$

Step 2

Solve for $a$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{4}$ and $a^{2}$ .

$\frac{a^{2}}{4} - \frac{1}{9} b^{2} = 0$

Step 2.1.2

Combine $b^{2}$ and $\frac{1}{9}$ .

$\frac{a^{2}}{4} - \frac{b^{2}}{9} = 0$

Step 2.2

Add $\frac{b^{2}}{9}$ to both sides of the equation.

$\frac{a^{2}}{4} = \frac{b^{2}}{9}$

Step 2.3

Multiply both sides of the equation by $4$ .

$4 \frac{a^{2}}{4} = 4 \frac{b^{2}}{9}$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $4$ .

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Step 2.4.1.1.1

Cancel the common factor.

$4 \frac{a^{2}}{4} = 4 \frac{b^{2}}{9}$

Step 2.4.1.1.2

Rewrite the expression.

$a^{2} = 4 \frac{b^{2}}{9}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Combine $4$ and $\frac{b^{2}}{9}$ .

$a^{2} = \frac{4 b^{2}}{9}$

Step 2.5

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

$a = \pm \sqrt{\frac{4 b^{2}}{9}}$

Step 2.6

Simplify $\pm \sqrt{\frac{4 b^{2}}{9}}$ .

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Step 2.6.1

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

$a = \pm \sqrt{\frac{{(2 b)}^{2}}{9}}$

Step 2.6.2

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

$a = \pm \sqrt{\frac{{(2 b)}^{2}}{3^{2}}}$

Step 2.6.3

Rewrite $\frac{{(2 b)}^{2}}{3^{2}}$ as ${(\frac{2 b}{3})}^{2}$ .

$a = \pm \sqrt{{(\frac{2 b}{3})}^{2}}$

Step 2.6.4

Pull terms out from under the radical, assuming positive real numbers.

$a = \pm \frac{2 b}{3}$

Step 2.7

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

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Step 2.7.1

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

$a = \frac{2 b}{3}$

Step 2.7.2

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

$a = - \frac{2 b}{3}$

Step 2.7.3

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

$a = \frac{2 b}{3}$

$a = - \frac{2 b}{3}$

$a = \frac{2 b}{3}$

$a = - \frac{2 b}{3}$

$a = \frac{2 b}{3}$

$a = - \frac{2 b}{3}$

Step 3

Set the denominator in $\frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$ equal to $0$ to find where the expression is undefined.

$\frac{3}{4} a + \frac{1}{2} b = 0$

Step 4

Solve for $a$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine $\frac{3}{4}$ and $a$ .

$\frac{3 a}{4} + \frac{1}{2} b = 0$

Step 4.1.2

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

$\frac{3 a}{4} + \frac{b}{2} = 0$

Step 4.2

Subtract $\frac{b}{2}$ from both sides of the equation.

$\frac{3 a}{4} = - \frac{b}{2}$

Step 4.3

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} (- \frac{b}{2})$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Simplify $\frac{4}{3} \cdot \frac{3 a}{4}$ .

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Step 4.4.1.1.1

Cancel the common factor of $4$ .

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Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} (- \frac{b}{2})$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 a) = \frac{4}{3} (- \frac{b}{2})$

Step 4.4.1.1.2

Cancel the common factor of $3$ .

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Step 4.4.1.1.2.1

Factor $3$ out of $3 a$ .

$\frac{1}{3} (3 (a)) = \frac{4}{3} (- \frac{b}{2})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 a) = \frac{4}{3} (- \frac{b}{2})$

Step 4.4.1.1.2.3

Rewrite the expression.

$a = \frac{4}{3} (- \frac{b}{2})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $\frac{4}{3} (- \frac{b}{2})$ .

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Step 4.4.2.1.1

Cancel the common factor of $2$ .

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Step 4.4.2.1.1.1

Move the leading negative in $- \frac{b}{2}$ into the numerator.

$a = \frac{4}{3} \cdot \frac{- b}{2}$

Step 4.4.2.1.1.2

Factor $2$ out of $4$ .

$a = \frac{2 (2)}{3} \cdot \frac{- b}{2}$

Step 4.4.2.1.1.3

Cancel the common factor.

$a = \frac{2 \cdot 2}{3} \cdot \frac{- b}{2}$

Step 4.4.2.1.1.4

Rewrite the expression.

$a = \frac{2}{3} (- b)$

Step 4.4.2.1.2

Combine $\frac{2}{3}$ and $b$ .

$a = - \frac{2 b}{3}$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${a | a \in ℝ}$