Solve for x (e^(x^2))/(e^2)=e^(-x)

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

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

Solve for $x$ .

Add $x$ to both sides of the equation.

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

Factor $x^{2} - 2 + x$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$(x - 1) (x + 2) = 0$

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

$x - 1 = 0$

$x + 2 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

Solve for x (e^(x^2))/(e^2)=e^(-x)

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