Precalculus Examples

$8^{x^{2} - 2 x} = \frac{1}{2}$

Step 1

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

$8^{x^{2} - 2 x} = \frac{1}{2^{1}}$

Step 2

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

$8^{x^{2} - 2 x} = 2^{- 1}$

Step 3

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

$2^{3 (x^{2} - 2 x)} = 2^{- 1}$

Step 4

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

$3 (x^{2} - 2 x) = - 1$

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $3 (x^{2} - 2 x) = - 1$ by $3$ and simplify.

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Step 5.1.1

Divide each term in $3 (x^{2} - 2 x) = - 1$ by $3$ .

$\frac{3 (x^{2} - 2 x)}{3} = \frac{- 1}{3}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $3$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} - 2 x)}{3} = \frac{- 1}{3}$

Step 5.1.2.1.2

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

$x^{2} - 2 x = \frac{- 1}{3}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Move the negative in front of the fraction.

$x^{2} - 2 x = - \frac{1}{3}$

Step 5.2

Add $\frac{1}{3}$ to both sides of the equation.

$x^{2} - 2 x + \frac{1}{3} = 0$

Step 5.3

Multiply through by the least common denominator $3$ , then simplify.

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Step 5.3.1

Apply the distributive property.

$3 x^{2} + 3 (- 2 x) + 3 (\frac{1}{3}) = 0$

Step 5.3.2

Simplify.

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Step 5.3.2.1

Multiply $- 2$ by $3$ .

$3 x^{2} - 6 x + 3 (\frac{1}{3}) = 0$

Step 5.3.2.2

Cancel the common factor of $3$ .

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Step 5.3.2.2.1

Cancel the common factor.

$3 x^{2} - 6 x + 3 (\frac{1}{3}) = 0$

Step 5.3.2.2.2

Rewrite the expression.

$3 x^{2} - 6 x + 1 = 0$

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

Substitute the values $a = 3$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 5.6.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 12 \cdot 1}}{2 \cdot 3}$

Step 5.6.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 12}}{2 \cdot 3}$

Step 5.6.1.3

Subtract $12$ from $36$ .

$x = \frac{6 \pm \sqrt{24}}{2 \cdot 3}$

Step 5.6.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 5.6.1.4.1

Factor $4$ out of $24$ .

$x = \frac{6 \pm \sqrt{4 (6)}}{2 \cdot 3}$

Step 5.6.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 3}$

Step 5.6.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{6}}{2 \cdot 3}$

Step 5.6.2

Multiply $2$ by $3$ .

$x = \frac{6 \pm 2 \sqrt{6}}{6}$

Step 5.6.3

Simplify $\frac{6 \pm 2 \sqrt{6}}{6}$ .

$x = \frac{3 \pm \sqrt{6}}{3}$

Step 5.7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{6}}{3}, \frac{3 - \sqrt{6}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{6}}{3}, \frac{3 - \sqrt{6}}{3}$

Decimal Form:

$x = 1.81649658 \dots, 0.18350341 \dots$