Precalculus Examples

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

Step 1

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

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

Step 2

Solve for $b$ .

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Step 2.1

Subtract $2 a^{2}$ from both sides of the equation.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Cancel the common factor of $a$ .

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Step 2.2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2.2

Divide $b$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $a^{2}$ and $a$ .

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Step 2.2.3.1.1

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

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

Step 2.2.3.1.2

Move the negative one from the denominator of $\frac{- 2 a}{- 1}$ .

$b = - 1 \cdot (- 2 a)$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Rewrite $- 1 \cdot (- 2 a)$ as $- (- 2 a)$ .

$b = - (- 2 a)$

Step 2.2.3.2.2

Multiply $- 2$ by $- 1$ .

$b = 2 a$

Step 3

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

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

Step 4

Solve for $b$ .

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Step 4.1

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

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Step 4.1.1

Factor $b$ out of $2 a b$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$b (2 a - b) = 0$

Step 4.2

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

$b = 0$

$2 a - b = 0$

Step 4.3

Set $b$ equal to $0$ .

$b = 0$

Step 4.4

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

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Step 4.4.1

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

$2 a - b = 0$

Step 4.4.2

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

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Step 4.4.2.1

Subtract $2 a$ from both sides of the equation.

$- b = - 2 a$

Step 4.4.2.2

Divide each term in $- b = - 2 a$ by $- 1$ and simplify.

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Step 4.4.2.2.1

Divide each term in $- b = - 2 a$ by $- 1$ .

$\frac{- b}{- 1} = \frac{- 2 a}{- 1}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{b}{1} = \frac{- 2 a}{- 1}$

Step 4.4.2.2.2.2

Divide $b$ by $1$ .

$b = \frac{- 2 a}{- 1}$

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative one from the denominator of $\frac{- 2 a}{- 1}$ .

$b = - 1 \cdot (- 2 a)$

Step 4.4.2.2.3.2

Rewrite $- 1 \cdot (- 2 a)$ as $- (- 2 a)$ .

$b = - (- 2 a)$

Step 4.4.2.2.3.3

Multiply $- 2$ by $- 1$ .

$b = 2 a$

Step 4.5

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

$b = 0$

$b = 2 a$

$b = 0$

$b = 2 a$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${b | b \neq 0}$