Precalculus Examples

$4^{x - x^{2}} = \frac{1}{2}$

Step 1

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

$4^{x - x^{2}} = \frac{1}{2^{1}}$

Step 2

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

$4^{x - x^{2}} = 2^{- 1}$

Step 3

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

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

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $2$ .

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Step 5.1.2.1.1

Cancel the common factor.

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

Step 5.1.2.1.2

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

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

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.

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

Step 5.2

Add $\frac{1}{2}$ to both sides of the equation.

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

Step 5.3

Multiply through by the least common denominator $2$ , then simplify.

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Simplify.

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Step 5.3.2.1

Multiply $- 1$ by $2$ .

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

Step 5.3.2.2

Cancel the common factor of $2$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

Step 5.3.3

Reorder $2 x$ and $- 2 x^{2}$ .

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

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

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

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 $2$ to the power of $2$ .

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

Step 5.6.1.2

Multiply $- 4 \cdot - 2 \cdot 1$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 2 \pm \sqrt{4 + 8 \cdot 1}}{2 \cdot - 2}$

Step 5.6.1.2.2

Multiply $8$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 + 8}}{2 \cdot - 2}$

Step 5.6.1.3

Add $4$ and $8$ .

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

Step 5.6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.6.1.4.1

Factor $4$ out of $12$ .

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

Step 5.6.1.4.2

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

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

Step 5.6.1.5

Pull terms out from under the radical.

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

Step 5.6.2

Multiply $2$ by $- 2$ .

$x = \frac{- 2 \pm 2 \sqrt{3}}{- 4}$

Step 5.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{3}}{- 4}$ .

$x = \frac{1 \pm \sqrt{3}}{2}$

Step 5.7

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Decimal Form:

$x = 1.36602540 \dots, - 0.36602540 \dots$