Algebra Examples

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

Step 1

Factor out $2^{x}$ from the expression.

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

Step 2

Simplify the expression.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$2^{x} (\frac{4}{4} + \frac{1}{4}) = 5$

Step 2.2

Combine the numerators over the common denominator.

$2^{x} (\frac{4 + 1}{4}) = 5$

Step 2.3

Add $4$ and $1$ .

$2^{x} (\frac{5}{4}) = 5$

Step 3

Combine $2^{x}$ and $\frac{5}{4}$ .

$\frac{2^{x} \cdot 5}{4} = 5$

Step 4

Move $5$ to the left of $2^{x}$ .

$\frac{5 \cdot 2^{x}}{4} = 5$

Step 5

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 \cdot 2^{x}}{4} = \frac{4}{5} \cdot 5$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{4}{5} \cdot \frac{5 \cdot 2^{x}}{4}$ .

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Step 6.1.1.1

Combine.

$\frac{4 (5 \cdot 2^{x})}{5 \cdot 4} = \frac{4}{5} \cdot 5$

Step 6.1.1.2

Cancel the common factor of $4$ .

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Step 6.1.1.2.1

Cancel the common factor.

$\frac{4 (5 \cdot 2^{x})}{5 \cdot 4} = \frac{4}{5} \cdot 5$

Step 6.1.1.2.2

Rewrite the expression.

$\frac{5 \cdot 2^{x}}{5} = \frac{4}{5} \cdot 5$

Step 6.1.1.3

Cancel the common factor of $5$ .

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Step 6.1.1.3.1

Cancel the common factor.

$\frac{5 \cdot 2^{x}}{5} = \frac{4}{5} \cdot 5$

Step 6.1.1.3.2

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

$2^{x} = \frac{4}{5} \cdot 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Cancel the common factor.

$2^{x} = \frac{4}{5} \cdot 5$

Step 6.2.1.2

Rewrite the expression.

$2^{x} = 4$

Step 7

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

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

Step 8

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

$x = 2$