Precalculus Examples

$\frac{c + 6 b}{a c + 2 b c - 6 a b - 3 a^{2}} + \frac{2 b}{a^{2} + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1

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

$a c + 2 b c - 6 a b - 3 a^{2} = 0$

Step 2

Solve for $c$ .

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Step 2.1

Move all terms not containing $c$ to the right side of the equation.

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Step 2.1.1

Add $6 a b$ to both sides of the equation.

$a c + 2 b c - 3 a^{2} = 6 a b$

Step 2.1.2

Add $3 a^{2}$ to both sides of the equation.

$a c + 2 b c = 6 a b + 3 a^{2}$

Step 2.2

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

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Step 2.2.1

Factor $c$ out of $a c$ .

$c a + 2 b c = 6 a b + 3 a^{2}$

Step 2.2.2

Factor $c$ out of $2 b c$ .

$c a + c (2 b) = 6 a b + 3 a^{2}$

Step 2.2.3

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

$c (a + 2 b) = 6 a b + 3 a^{2}$

Step 2.3

Divide each term in $c (a + 2 b) = 6 a b + 3 a^{2}$ by $a + 2 b$ and simplify.

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Step 2.3.1

Divide each term in $c (a + 2 b) = 6 a b + 3 a^{2}$ by $a + 2 b$ .

$\frac{c (a + 2 b)}{a + 2 b} = \frac{6 a b}{a + 2 b} + \frac{3 a^{2}}{a + 2 b}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $a + 2 b$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{c (a + 2 b)}{a + 2 b} = \frac{6 a b}{a + 2 b} + \frac{3 a^{2}}{a + 2 b}$

Step 2.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{6 a b}{a + 2 b} + \frac{3 a^{2}}{a + 2 b}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Combine the numerators over the common denominator.

$c = \frac{6 a b + 3 a^{2}}{a + 2 b}$

Step 2.3.3.2

Factor $3 a$ out of $6 a b + 3 a^{2}$ .

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Step 2.3.3.2.1

Factor $3 a$ out of $6 a b$ .

$c = \frac{3 a (2 b) + 3 a^{2}}{a + 2 b}$

Step 2.3.3.2.2

Factor $3 a$ out of $3 a^{2}$ .

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

Step 2.3.3.2.3

Factor $3 a$ out of $3 a (2 b) + 3 a (a)$ .

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

Step 2.3.3.3

Cancel the common factor of $2 b + a$ and $a + 2 b$ .

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Step 2.3.3.3.1

Reorder terms.

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

Step 2.3.3.3.2

Cancel the common factor.

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

Step 2.3.3.3.3

Divide $3 a$ by $1$ .

$c = 3 a$

Step 3

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

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

Step 4

Solve for $c$ .

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Step 4.1

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

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Step 4.1.1

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

$a \cdot a + 2 a b = 0$

Step 4.1.2

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

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

Step 4.1.3

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

$a (a + 2 b) = 0$

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 $0$ .

$a = 0$

$a + 2 b = 0$

Step 4.3

Set $a$ equal to $0$ .

$a = 0$

Step 4.4

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

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Step 4.4.1

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

$a + 2 b = 0$

Step 4.4.2

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

$a = - 2 b$

Step 4.5

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

$a = 0$

$a = - 2 b$

$a = 0$

$a = - 2 b$

Step 5

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

$a c - 3 a^{2} = 0$

Step 6

Solve for $c$ .

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Step 6.1

Add $3 a^{2}$ to both sides of the equation.

$a c = 3 a^{2}$

Step 6.2

Divide each term in $a c = 3 a^{2}$ by $a$ and simplify.

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Step 6.2.1

Divide each term in $a c = 3 a^{2}$ by $a$ .

$\frac{a c}{a} = \frac{3 a^{2}}{a}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $a$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{a c}{a} = \frac{3 a^{2}}{a}$

Step 6.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{3 a^{2}}{a}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

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Step 6.2.3.1.1

Factor $a$ out of $3 a^{2}$ .

$c = \frac{a (3 a)}{a}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Raise $a$ to the power of $1$ .

$c = \frac{a (3 a)}{a^{1}}$

Step 6.2.3.1.2.2

Factor $a$ out of $a^{1}$ .

$c = \frac{a (3 a)}{a \cdot 1}$

Step 6.2.3.1.2.3

Cancel the common factor.

$c = \frac{a (3 a)}{a \cdot 1}$

Step 6.2.3.1.2.4

Rewrite the expression.

$c = \frac{3 a}{1}$

Step 6.2.3.1.2.5

Divide $3 a$ by $1$ .

$c = 3 a$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${c | c \neq 0}$