Calculus Examples

$3^{x - x^{2}} = \frac{1}{9^{x}}$

Step 1

Move $9^{x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{x - x^{2}} = 9^{- x}$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

Multiply $- 1$ by $2$ .

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

Step 4.2

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

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Step 4.2.1

Add $2 x$ to both sides of the equation.

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

Step 4.2.2

Add $x$ and $2 x$ .

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

Step 4.3

Factor $- x$ out of $- x^{2} + 3 x$ .

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Step 4.3.1

Factor $- x$ out of $- x^{2}$ .

$- x \cdot x + 3 x = 0$

Step 4.3.2

Factor $- x$ out of $3 x$ .

$- x \cdot x - x \cdot - 3 = 0$

Step 4.3.3

Factor $- x$ out of $- x (x) - x (- 3)$ .

$- x (x - 3) = 0$

Step 4.4

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

$x = 0$

$x - 3 = 0$

Step 4.5

Set $x$ equal to $0$ .

$x = 0$

Step 4.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.7

The final solution is all the values that make $- x (x - 3) = 0$ true.

$x = 0, 3$