Precalculus Examples

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

Step 1

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

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

Step 2

Solve for $a$ .

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Step 2.1

Set the numerator equal to zero.

$4 a^{3} - a b^{2} = 0$

Step 2.2

Solve the equation for $a$ .

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Step 2.2.1

Factor $a$ out of $4 a^{3} - a b^{2}$ .

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Step 2.2.1.1

Factor $a$ out of $4 a^{3}$ .

$a (4 a^{2}) - a b^{2} = 0$

Step 2.2.1.2

Factor $a$ out of $- a b^{2}$ .

$a (4 a^{2}) + a (- 1 b^{2}) = 0$

Step 2.2.1.3

Factor $a$ out of $a (4 a^{2}) + a (- 1 b^{2})$ .

$a (4 a^{2} - 1 b^{2}) = 0$

Step 2.2.2

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

$a ({(2 a)}^{2} - 1 b^{2}) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a$ and $b = b$ .

$a ((2 a + b) (2 a - b)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$a (2 a + b) (2 a - b) = 0$

Step 2.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a = 0$

$2 a + b = 0$

$2 a - b = 0$

Step 2.2.5

Set $a$ equal to $0$ .

$a = 0$

Step 2.2.6

Set $2 a + b$ equal to $0$ and solve for $a$ .

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Step 2.2.6.1

Set $2 a + b$ equal to $0$ .

$2 a + b = 0$

Step 2.2.6.2

Solve $2 a + b = 0$ for $a$ .

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Step 2.2.6.2.1

Subtract $b$ from both sides of the equation.

$2 a = - b$

Step 2.2.6.2.2

Divide each term in $2 a = - b$ by $2$ and simplify.

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Step 2.2.6.2.2.1

Divide each term in $2 a = - b$ by $2$ .

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

Step 2.2.6.2.2.2

Simplify the left side.

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Step 2.2.6.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.6.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.6.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.2.6.2.2.3

Simplify the right side.

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Step 2.2.6.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.7

Set $2 a - b$ equal to $0$ and solve for $a$ .

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Step 2.2.7.1

Set $2 a - b$ equal to $0$ .

$2 a - b = 0$

Step 2.2.7.2

Solve $2 a - b = 0$ for $a$ .

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Step 2.2.7.2.1

Add $b$ to both sides of the equation.

$2 a = b$

Step 2.2.7.2.2

Divide each term in $2 a = b$ by $2$ and simplify.

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Step 2.2.7.2.2.1

Divide each term in $2 a = b$ by $2$ .

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

Step 2.2.7.2.2.2

Simplify the left side.

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Step 2.2.7.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.7.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.7.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.2.8

The final solution is all the values that make $a (2 a + b) (2 a - b) = 0$ true.

$a = 0$

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

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

$a = 0$

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

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

$a = 0$

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

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

Step 3

The domain is all values of $a$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${a | a \neq 0}$