Algebra Examples

$2^{x} + 2^{- x} = \frac{5}{2}$

Step 1

Rewrite $2^{- x}$ as exponentiation.

$2^{x} + {(2^{x})}^{- 1} = \frac{5}{2}$

Step 2

Substitute $u$ for $2^{x}$ .

$u + u^{- 1} = \frac{5}{2}$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$u + \frac{1}{u} = \frac{5}{2}$

Step 4

Solve for $u$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

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

$1, u, 2$

Step 4.1.2

Since $1, u, 2$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 2$ then find LCM for the variable part $u^{1}$ .

Step 4.1.3

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 4.1.4

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

Not prime

Step 4.1.5

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

$2$ is a prime number

Step 4.1.6

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 4.1.7

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

$u^{1} = u$

$u$ occurs $1$ time.

Step 4.1.8

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

$u$

Step 4.1.9

The LCM for $1, u, 2$ is the numeric part $2$ multiplied by the variable part.

$2 u$

Step 4.2

Multiply each term in $u + \frac{1}{u} = \frac{5}{2}$ by $2 u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $u + \frac{1}{u} = \frac{5}{2}$ by $2 u$ .

$u (2 u) + \frac{1}{u} (2 u) = \frac{5}{2} (2 u)$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Rewrite using the commutative property of multiplication.

$2 u \cdot u + \frac{1}{u} (2 u) = \frac{5}{2} (2 u)$

Step 4.2.2.1.2

Multiply $u$ by $u$ by adding the exponents.

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Step 4.2.2.1.2.1

Move $u$ .

$2 (u \cdot u) + \frac{1}{u} (2 u) = \frac{5}{2} (2 u)$

Step 4.2.2.1.2.2

Multiply $u$ by $u$ .

$2 u^{2} + \frac{1}{u} (2 u) = \frac{5}{2} (2 u)$

Step 4.2.2.1.3

Rewrite using the commutative property of multiplication.

$2 u^{2} + 2 \frac{1}{u} u = \frac{5}{2} (2 u)$

Step 4.2.2.1.4

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

$2 u^{2} + \frac{2}{u} u = \frac{5}{2} (2 u)$

Step 4.2.2.1.5

Cancel the common factor of $u$ .

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Step 4.2.2.1.5.1

Cancel the common factor.

$2 u^{2} + \frac{2}{u} u = \frac{5}{2} (2 u)$

Step 4.2.2.1.5.2

Rewrite the expression.

$2 u^{2} + 2 = \frac{5}{2} (2 u)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $2$ .

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Step 4.2.3.1.1

Factor $2$ out of $2 u$ .

$2 u^{2} + 2 = \frac{5}{2} (2 (u))$

Step 4.2.3.1.2

Cancel the common factor.

$2 u^{2} + 2 = \frac{5}{2} (2 u)$

Step 4.2.3.1.3

Rewrite the expression.

$2 u^{2} + 2 = 5 u$

Step 4.3

Solve the equation.

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Step 4.3.1

Subtract $5 u$ from both sides of the equation.

$2 u^{2} + 2 - 5 u = 0$

Step 4.3.2

Factor by grouping.

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Step 4.3.2.1

Reorder terms.

$2 u^{2} - 5 u + 2 = 0$

Step 4.3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

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Step 4.3.2.2.1

Factor $- 5$ out of $- 5 u$ .

$2 u^{2} - 5 u + 2 = 0$

Step 4.3.2.2.2

Rewrite $- 5$ as $- 1$ plus $- 4$

$2 u^{2} + (- 1 - 4) u + 2 = 0$

Step 4.3.2.2.3

Apply the distributive property.

$2 u^{2} - 1 u - 4 u + 2 = 0$

Step 4.3.2.3

Factor out the greatest common factor from each group.

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Step 4.3.2.3.1

Group the first two terms and the last two terms.

$(2 u^{2} - 1 u) - 4 u + 2 = 0$

Step 4.3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$u (2 u - 1) - 2 (2 u - 1) = 0$

Step 4.3.2.4

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(2 u - 1) (u - 2) = 0$

Step 4.3.3

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

$2 u - 1 = 0$

$u - 2 = 0$

Step 4.3.4

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 4.3.4.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 4.3.4.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 4.3.4.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 4.3.4.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 4.3.4.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 4.3.4.2.2.2

Simplify the left side.

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Step 4.3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 4.3.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 4.3.5

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 4.3.5.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 4.3.5.2

Add $2$ to both sides of the equation.

$u = 2$

Step 4.3.6

The final solution is all the values that make $(2 u - 1) (u - 2) = 0$ true.

$u = \frac{1}{2}, 2$

Step 5

Substitute $\frac{1}{2}$ for $u$ in $u = 2^{x}$ .

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

Step 6

Solve $\frac{1}{2} = 2^{x}$ .

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Step 6.1

Rewrite the equation as $2^{x} = \frac{1}{2}$ .

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

Step 6.2

Raise $2$ to the power of $1$ .

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

Step 6.3

Move $2^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2^{x} = 2^{- 1}$

Step 6.4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = - 1$

Step 7

Substitute $2$ for $u$ in $u = 2^{x}$ .

$2 = 2^{x}$

Step 8

Solve $2 = 2^{x}$ .

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Step 8.1

Rewrite the equation as $2^{x} = 2$ .

$2^{x} = 2$

Step 8.2

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{1}$

Step 8.3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = 1$

Step 9

List the solutions that makes the equation true.

$x = - 1, 1$