Precalculus Examples

$\frac{1}{x} + \frac{1}{x + 2} = \frac{1}{2}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$x, x + 2, 2$

Step 1.2

Since $x, x + 2, 2$ 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 $x, x + 2, 2$ are:

1. Find the LCM for the numeric part $1, 1, 2$ .

2. Find the LCM for the variable part $x^{1}$ .

3. Find the LCM for the compound variable part $x + 2$ .

4. Multiply each LCM together.

Step 1.3

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 1.4

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

Not prime

Step 1.5

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 1.6

The LCM of $1, 1, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 1.7

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

$x^{1} = x$

$x$ occurs $1$ time.

Step 1.8

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

$x$

Step 1.9

The factor for $x + 2$ is $x + 2$ itself.

$(x + 2) = x + 2$

$(x + 2)$ occurs $1$ time.

Step 1.10

The LCM of $x + 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x + 2$

Step 1.11

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

$2 x (x + 2)$

Step 2

Multiply each term in $\frac{1}{x} + \frac{1}{x + 2} = \frac{1}{2}$ by $2 x (x + 2)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{1}{x} + \frac{1}{x + 2} = \frac{1}{2}$ by $2 x (x + 2)$ .

$\frac{1}{x} (2 x (x + 2)) + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{1}{x} (x (x + 2)) + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.2

Combine $2$ and $\frac{1}{x}$ .

$\frac{2}{x} (x (x + 2)) + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.3

Cancel the common factor of $x$ .

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Step 2.2.1.3.1

Cancel the common factor.

$\frac{2}{x} (x (x + 2)) + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.3.2

Rewrite the expression.

$2 (x + 2) + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.4

Apply the distributive property.

$2 x + 2 \cdot 2 + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.5

Multiply $2$ by $2$ .

$2 x + 4 + \frac{1}{x + 2} (2 x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.6

Rewrite using the commutative property of multiplication.

$2 x + 4 + 2 \frac{1}{x + 2} (x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.7

Combine $2$ and $\frac{1}{x + 2}$ .

$2 x + 4 + \frac{2}{x + 2} (x (x + 2)) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.8

Cancel the common factor of $x + 2$ .

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Step 2.2.1.8.1

Factor $x + 2$ out of $x (x + 2)$ .

$2 x + 4 + \frac{2}{x + 2} ((x + 2) x) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.8.2

Cancel the common factor.

$2 x + 4 + \frac{2}{x + 2} ((x + 2) x) = \frac{1}{2} (2 x (x + 2))$

Step 2.2.1.8.3

Rewrite the expression.

$2 x + 4 + 2 x = \frac{1}{2} (2 x (x + 2))$

Step 2.2.2

Add $2 x$ and $2 x$ .

$4 x + 4 = \frac{1}{2} (2 x (x + 2))$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $2$ .

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Step 2.3.1.1

Factor $2$ out of $2 x (x + 2)$ .

$4 x + 4 = \frac{1}{2} (2 (x (x + 2)))$

Step 2.3.1.2

Cancel the common factor.

$4 x + 4 = \frac{1}{2} (2 (x (x + 2)))$

Step 2.3.1.3

Rewrite the expression.

$4 x + 4 = x (x + 2)$

Step 2.3.2

Apply the distributive property.

$4 x + 4 = x \cdot x + x \cdot 2$

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

Multiply $x$ by $x$ .

$4 x + 4 = x^{2} + x \cdot 2$

Step 2.3.3.2

Move $2$ to the left of $x$ .

$4 x + 4 = x^{2} + 2 x$

Step 3

Solve the equation.

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 2 x = 4 x + 4$

Step 3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.1

Subtract $4 x$ from both sides of the equation.

$x^{2} + 2 x - 4 x = 4$

Step 3.2.2

Subtract $4 x$ from $2 x$ .

$x^{2} - 2 x = 4$

Step 3.3

Subtract $4$ from both sides of the equation.

$x^{2} - 2 x - 4 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.6.1.2

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

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 3.6.1.3

Add $4$ and $16$ .

$x = \frac{2 \pm \sqrt{20}}{2 \cdot 1}$

Step 3.6.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.6.1.4.1

Factor $4$ out of $20$ .

$x = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 3.6.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{5}}{2}$

Step 3.6.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{2}$ .

$x = 1 \pm \sqrt{5}$

Step 3.7

The final answer is the combination of both solutions.

$x = 1 + \sqrt{5}, 1 - \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{5}, 1 - \sqrt{5}$

Decimal Form:

$x = 3.23606797 \dots, - 1.23606797 \dots$