Precalculus Examples

$\frac{2 a + b}{2 a^{2} - a b} - \frac{16 a}{4 a^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1

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

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

Step 2

Solve for $a$ .

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $a$ out of $- a b$ .

$a (2 a) + a (- 1 b) = 0$

Step 2.1.3

Factor $a$ out of $a (2 a) + a (- 1 b)$ .

$a (2 a - 1 b) = 0$

Step 2.2

Rewrite $- 1 b$ as $- b$ .

$a (2 a - b) = 0$

Step 2.3

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

$a = 0$

$2 a - b = 0$

Step 2.4

Set $a$ equal to $0$ .

$a = 0$

Step 2.5

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

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Step 2.5.1

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

$2 a - b = 0$

Step 2.5.2

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

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Step 2.5.2.1

Add $b$ to both sides of the equation.

$2 a = b$

Step 2.5.2.2

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

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

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

Step 2.5.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.6

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

$a = 0$

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

$a = 0$

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

Step 3

Set the denominator in $\frac{16 a}{4 a^{2} - b^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 4

Solve for $a$ .

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Step 4.1

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

$4 a^{2} = b^{2}$

Step 4.2

Divide each term in $4 a^{2} = b^{2}$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 a^{2} = b^{2}$ by $4$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

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

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

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

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

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

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

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

Step 5

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

$2 a^{2} + a b = 0$

Step 6

Solve for $a$ .

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Factor $a$ out of $a b$ .

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

Step 6.1.3

Factor $a$ out of $a (2 a) + a (b)$ .

$a (2 a + b) = 0$

Step 6.2

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

$a = 0$

$2 a + b = 0$

Step 6.3

Set $a$ equal to $0$ .

$a = 0$

Step 6.4

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

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Step 6.4.1

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

$2 a + b = 0$

Step 6.4.2

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

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Step 6.4.2.1

Subtract $b$ from both sides of the equation.

$2 a = - b$

Step 6.4.2.2

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

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Step 6.4.2.2.1

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

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

Step 6.4.2.2.2

Simplify the left side.

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Step 6.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.4.2.2.2.1.1

Cancel the common factor.

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

Step 6.4.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 6.4.2.2.3

Simplify the right side.

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Step 6.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 6.5

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

$a = 0$

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

$a = 0$

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

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${a | a \neq 0}$