Precalculus Examples

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

Step 1

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

$a + b = 0$

Step 2

Subtract $a$ from both sides of the equation.

$b = - a$

Step 3

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

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

Step 4

Solve for $b$ .

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Step 4.1

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

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Step 4.1.1

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

$a \cdot a - a c = 0$

Step 4.1.2

Factor $a$ out of $- a c$ .

$a \cdot a + a (- 1 c) = 0$

Step 4.1.3

Factor $a$ out of $a \cdot a + a (- 1 c)$ .

$a (a - 1 c) = 0$

Step 4.2

Rewrite $- 1 c$ as $- c$ .

$a (a - c) = 0$

Step 4.3

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

$a = 0$

$a - c = 0$

Step 4.4

Set $a$ equal to $0$ .

$a = 0$

Step 4.5

Set $a - c$ equal to $0$ and solve for $a$ .

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Step 4.5.1

Set $a - c$ equal to $0$ .

$a - c = 0$

Step 4.5.2

Add $c$ to both sides of the equation.

$a = c$

Step 4.6

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

$a = 0$

$a = c$

$a = 0$

$a = c$

Step 5

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

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

Step 6

Solve for $b$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

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

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Dividing two negative values results in a positive value.

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

Step 6.2.3.2

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

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

Step 6.3

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| b | = | a |$

Step 6.4

Solve for $b$ .

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Step 6.4.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$b = \pm | a |$

Step 6.4.2

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

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Step 6.4.2.1

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

$b = | a |$

Step 6.4.2.2

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

$b = - | a |$

Step 6.4.2.3

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

$b = | a |$

$b = - | a |$

$b = | a |$

$b = - | a |$

$b = | a |$

$b = - | a |$

$b = | a |$

$b = - | a |$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${b | b \neq 0}$