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

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

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

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

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

$x = x^{2} - 2$

Solve for $x$ .

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Subtract $x^{2}$ from both sides of the equation.

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

Move $2$ to the left side of the equation by adding it to both sides.

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

Factor the left side of the equation.

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Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u - u^{2} + 2$

Factor by grouping.

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Reorder terms.

$- u^{2} + u + 2$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = 1$ .

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Multiply by $1$ .

$- u^{2} + 1 u + 2$

Rewrite $1$ as $- 1$ plus $2$

$- u^{2} + (- 1 + 2) u + 2$

Apply the distributive property.

$- u^{2} - 1 u + 2 u + 2$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$(- u^{2} - 1 u) + 2 u + 2$

Factor out the greatest common factor (GCF) from each group.

$u (- u - 1) - 2 (- u - 1)$

Factor the polynomial by factoring out the greatest common factor, $- u - 1$ .

$(- u - 1) (u - 2)$

Replace all occurrences of $u$ with $x$ .

$(- x - 1) (x - 2)$

Replace the left side with the factored expression.

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

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 the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$- x - 1 = 0$

Add $1$ to both sides of the equation.

$- x = 1$

Multiply each term in $- x = 1$ by $- 1$

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Multiply each term in $- x = 1$ by $- 1$ .

$(- x) \cdot - 1 = 1 \cdot - 1$

Multiply $(- x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 x = 1 \cdot - 1$

Multiply $x$ by $1$ .

$x = 1 \cdot - 1$

Multiply $- 1$ by $1$ .

$x = - 1$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x - 2 = 0$

Add $2$ to 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=e^(x^2-2)

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