Solve for x e^(x^2)=e^(3x)*1/(e^2)

$e^{x^{2}} = e^{3 x} \cdot \frac{1}{e^{2}}$

Step 1

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

$e^{x^{2}} = e^{3 x} \cdot e^{- 2}$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $x$ .

Subtract $3 x$ from both sides of the equation.

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

Add $2$ to both sides of the equation.

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

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

$- 2, - 1$

Write the factored form using these integers.

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

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

$x - 2 = 0$

$x - 1 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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.

$x = 1$

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

$x = 2, 1$

Solve for x e^(x^2)=e^(3x)*1/(e^2)

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