Calculus Examples

$e^{- x^{2}} = e^{x^{2} - 12 x}$

Step 1

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

$e^{- x^{2}} = e^{x^{2} - 12 x}$

Step 2

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

$- x^{2} = x^{2} - 12 x$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 12 x = - x^{2}$

Step 3.2

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

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Step 3.2.1

Add $x^{2}$ to both sides of the equation.

$x^{2} - 12 x + x^{2} = 0$

Step 3.2.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 12 x = 0$

Step 3.3

Factor $2 x$ out of $2 x^{2} - 12 x$ .

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Step 3.3.1

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

$2 x (x) - 12 x = 0$

Step 3.3.2

Factor $2 x$ out of $- 12 x$ .

$2 x (x) + 2 x (- 6) = 0$

Step 3.3.3

Factor $2 x$ out of $2 x (x) + 2 x (- 6)$ .

$2 x (x - 6) = 0$

Step 3.4

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

$x = 0$

$x - 6 = 0$

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

Step 3.6

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.6.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.7

The final solution is all the values that make $2 x (x - 6) = 0$ true.

$x = 0, 6$