Algebra Examples

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

Step 1

Add $\frac{1}{b}$ to both sides of the equation.

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a, a - b, b$

Step 2.2

Since $a, a - b, b$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $a, a - b, b$ are:

1. Find the LCM for the numeric part $1, 1, 1$ .

2. Find the LCM for the variable part $a^{1}, b^{1}$ .

3. Find the LCM for the compound variable part $a - b$ .

4. Multiply each LCM together.

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factor for $a^{1}$ is $a$ itself.

$a^{1} = a$

$a$ occurs $1$ time.

Step 2.7

The factor for $b^{1}$ is $b$ itself.

$b^{1} = b$

$b$ occurs $1$ time.

Step 2.8

The LCM of $a^{1}, b^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$a \cdot b$

Step 2.9

Multiply $a$ by $b$ .

$a b$

Step 2.10

The factor for $a - b$ is $a - b$ itself.

$(a - b) = a - b$

$(a - b)$ occurs $1$ time.

Step 2.11

The LCM of $a - b$ is the result of multiplying all factors the greatest number of times they occur in either term.

$a - b$

Step 2.12

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$a b (a - b)$

Step 3

Multiply each term in $\frac{1}{a} = \frac{1}{a - b} + \frac{1}{b}$ by $a b (a - b)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{a} = \frac{1}{a - b} + \frac{1}{b}$ by $a b (a - b)$ .

$\frac{1}{a} (a b (a - b)) = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $a$ .

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Step 3.2.1.1

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

$\frac{1}{a} (a (b (a - b))) = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2.1.2

Cancel the common factor.

$\frac{1}{a} (a (b (a - b))) = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2.1.3

Rewrite the expression.

$b (a - b) = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2.2

Apply the distributive property.

$b a + b (- b) = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$b a - b \cdot b = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2.4

Multiply $b$ by $b$ by adding the exponents.

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Step 3.2.4.1

Move $b$ .

$b a - (b \cdot b) = \frac{1}{a - b} (a b (a - b)) + \frac{1}{b} (a b (a - b))$

Step 3.2.4.2

Multiply $b$ by $b$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $a - b$ .

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

Cancel the common factor.

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

Step 3.3.1.1.3

Rewrite the expression.

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

Step 3.3.1.2

Cancel the common factor of $b$ .

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Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

Cancel the common factor.

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

Step 3.3.1.2.3

Rewrite the expression.

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

Step 3.3.1.3

Apply the distributive property.

$b a - b^{2} = a b + a \cdot a + a (- b)$

Step 3.3.1.4

Multiply $a$ by $a$ .

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

Step 3.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

Combine the opposite terms in $a b + a^{2} - a b$ .

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Step 3.3.2.1

Subtract $a b$ from $a b$ .

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

Step 3.3.2.2

Add $a^{2}$ and $0$ .

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

Step 4

Solve the equation.

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Step 4.1

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

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

Step 4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3

Substitute the values $a = - 1$ , $b = b$ , and $c = - b^{2}$ into the quadratic formula and solve for $a$ .

$\frac{- b \pm \sqrt{b^{2} - 4 \cdot (- 1 \cdot (- b^{2}))}}{2 \cdot - 1}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Factor $b^{2}$ out of $b^{2} - 4 \cdot - 1 \cdot - 1 b^{2}$ .

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Step 4.4.1.1.1

Multiply by $1$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot 1 - 4 \cdot - 1 \cdot (- 1 b^{2})}}{2 \cdot - 1}$

Step 4.4.1.1.2

Factor $b^{2}$ out of $- 4 \cdot - 1 \cdot - 1 b^{2}$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot 1 + b^{2} (- 4 \cdot - 1 \cdot - 1)}}{2 \cdot - 1}$

Step 4.4.1.1.3

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

$a = \frac{- b \pm \sqrt{b^{2} (1 - 4 \cdot - 1 \cdot - 1)}}{2 \cdot - 1}$

Step 4.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 1$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $- 1$ .

$a = \frac{- b \pm \sqrt{b^{2} (1 + 4 \cdot - 1)}}{2 \cdot - 1}$

Step 4.4.1.2.2

Multiply $4$ by $- 1$ .

$a = \frac{- b \pm \sqrt{b^{2} (1 - 4)}}{2 \cdot - 1}$

Step 4.4.1.3

Subtract $4$ from $1$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot - 3}}{2 \cdot - 1}$

Step 4.4.1.4

Pull terms out from under the radical.

$a = \frac{- b \pm b \sqrt{- 3}}{2 \cdot - 1}$

Step 4.4.1.5

Rewrite $- 3$ as $- 1 (3)$ .

$a = \frac{- b \pm b \sqrt{- 1 \cdot 3}}{2 \cdot - 1}$

Step 4.4.1.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$a = \frac{- b \pm b (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot - 1}$

Step 4.4.1.7

Rewrite $\sqrt{- 1}$ as $i$ .

$a = \frac{- b \pm b i \sqrt{3}}{2 \cdot - 1}$

Step 4.4.2

Multiply $2$ by $- 1$ .

$a = \frac{- b \pm b i \sqrt{3}}{- 2}$

Step 4.4.3

Simplify $\frac{- b \pm b i \sqrt{3}}{- 2}$ .

$a = \frac{b \pm b i \sqrt{3}}{2}$

Step 4.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Factor $b^{2}$ out of $b^{2} - 4 \cdot - 1 \cdot - 1 b^{2}$ .

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Step 4.5.1.1.1

Multiply by $1$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot 1 - 4 \cdot - 1 \cdot (- 1 b^{2})}}{2 \cdot - 1}$

Step 4.5.1.1.2

Factor $b^{2}$ out of $- 4 \cdot - 1 \cdot - 1 b^{2}$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot 1 + b^{2} (- 4 \cdot - 1 \cdot - 1)}}{2 \cdot - 1}$

Step 4.5.1.1.3

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

$a = \frac{- b \pm \sqrt{b^{2} (1 - 4 \cdot - 1 \cdot - 1)}}{2 \cdot - 1}$

Step 4.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 1$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $- 1$ .

$a = \frac{- b \pm \sqrt{b^{2} (1 + 4 \cdot - 1)}}{2 \cdot - 1}$

Step 4.5.1.2.2

Multiply $4$ by $- 1$ .

$a = \frac{- b \pm \sqrt{b^{2} (1 - 4)}}{2 \cdot - 1}$

Step 4.5.1.3

Subtract $4$ from $1$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot - 3}}{2 \cdot - 1}$

Step 4.5.1.4

Pull terms out from under the radical.

$a = \frac{- b \pm b \sqrt{- 3}}{2 \cdot - 1}$

Step 4.5.1.5

Rewrite $- 3$ as $- 1 (3)$ .

$a = \frac{- b \pm b \sqrt{- 1 \cdot 3}}{2 \cdot - 1}$

Step 4.5.1.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$a = \frac{- b \pm b (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot - 1}$

Step 4.5.1.7

Rewrite $\sqrt{- 1}$ as $i$ .

$a = \frac{- b \pm b i \sqrt{3}}{2 \cdot - 1}$

Step 4.5.2

Multiply $2$ by $- 1$ .

$a = \frac{- b \pm b i \sqrt{3}}{- 2}$

Step 4.5.3

Simplify $\frac{- b \pm b i \sqrt{3}}{- 2}$ .

$a = \frac{b \pm b i \sqrt{3}}{2}$

Step 4.5.4

Change the $\pm$ to $+$ .

$a = \frac{b + b i \sqrt{3}}{2}$

Step 4.5.5

Factor $b$ out of $b + b i \sqrt{3}$ .

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Step 4.5.5.1

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

$a = \frac{b + b i \sqrt{3}}{2}$

Step 4.5.5.2

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

$a = \frac{b \cdot 1 + b i \sqrt{3}}{2}$

Step 4.5.5.3

Factor $b$ out of $b i \sqrt{3}$ .

$a = \frac{b \cdot 1 + b (i \sqrt{3})}{2}$

Step 4.5.5.4

Factor $b$ out of $b \cdot 1 + b (i \sqrt{3})$ .

$a = \frac{b (1 + i \sqrt{3})}{2}$

Step 4.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Factor $b^{2}$ out of $b^{2} - 4 \cdot - 1 \cdot - 1 b^{2}$ .

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Step 4.6.1.1.1

Multiply by $1$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot 1 - 4 \cdot - 1 \cdot (- 1 b^{2})}}{2 \cdot - 1}$

Step 4.6.1.1.2

Factor $b^{2}$ out of $- 4 \cdot - 1 \cdot - 1 b^{2}$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot 1 + b^{2} (- 4 \cdot - 1 \cdot - 1)}}{2 \cdot - 1}$

Step 4.6.1.1.3

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

$a = \frac{- b \pm \sqrt{b^{2} (1 - 4 \cdot - 1 \cdot - 1)}}{2 \cdot - 1}$

Step 4.6.1.2

Multiply $- 4 \cdot - 1 \cdot - 1$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $- 1$ .

$a = \frac{- b \pm \sqrt{b^{2} (1 + 4 \cdot - 1)}}{2 \cdot - 1}$

Step 4.6.1.2.2

Multiply $4$ by $- 1$ .

$a = \frac{- b \pm \sqrt{b^{2} (1 - 4)}}{2 \cdot - 1}$

Step 4.6.1.3

Subtract $4$ from $1$ .

$a = \frac{- b \pm \sqrt{b^{2} \cdot - 3}}{2 \cdot - 1}$

Step 4.6.1.4

Pull terms out from under the radical.

$a = \frac{- b \pm b \sqrt{- 3}}{2 \cdot - 1}$

Step 4.6.1.5

Rewrite $- 3$ as $- 1 (3)$ .

$a = \frac{- b \pm b \sqrt{- 1 \cdot 3}}{2 \cdot - 1}$

Step 4.6.1.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$a = \frac{- b \pm b (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot - 1}$

Step 4.6.1.7

Rewrite $\sqrt{- 1}$ as $i$ .

$a = \frac{- b \pm b i \sqrt{3}}{2 \cdot - 1}$

Step 4.6.2

Multiply $2$ by $- 1$ .

$a = \frac{- b \pm b i \sqrt{3}}{- 2}$

Step 4.6.3

Simplify $\frac{- b \pm b i \sqrt{3}}{- 2}$ .

$a = \frac{b \pm b i \sqrt{3}}{2}$

Step 4.6.4

Change the $\pm$ to $-$ .

$a = \frac{b - b i \sqrt{3}}{2}$

Step 4.6.5

Simplify the numerator.

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Step 4.6.5.1

Factor $b$ out of $b - b i \sqrt{3}$ .

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Step 4.6.5.1.1

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

$a = \frac{b - b i \sqrt{3}}{2}$

Step 4.6.5.1.2

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

$a = \frac{b \cdot 1 - b i \sqrt{3}}{2}$

Step 4.6.5.1.3

Factor $b$ out of $- b i \sqrt{3}$ .

$a = \frac{b \cdot 1 + b (- 1 i \sqrt{3})}{2}$

Step 4.6.5.1.4

Factor $b$ out of $b \cdot 1 + b (- 1 i \sqrt{3})$ .

$a = \frac{b (1 - 1 i \sqrt{3})}{2}$

Step 4.6.5.2

Rewrite $- 1 i$ as $- i$ .

$a = \frac{b (1 - i \sqrt{3})}{2}$

Step 4.7

The final answer is the combination of both solutions.

$a = \frac{b (1 + i \sqrt{3})}{2}$

$a = \frac{b (1 - i \sqrt{3})}{2}$

$a = \frac{b (1 + i \sqrt{3})}{2}$

$a = \frac{b (1 - i \sqrt{3})}{2}$