Solve for x e^(x^2)=(e^x)*e^12

$e^{x^{2}} = (e^{x}) \cdot e^{12}$

Step 1

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

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

Step 2

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

$x^{2} = x + 12$

Step 3

Solve for $x$ .

Subtract $x$ from both sides of the equation.

$x^{2} - x = 12$

Subtract $12$ from both sides of the equation.

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

Factor $x^{2} - x - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$(x - 4) (x + 3) = 0$

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

$x - 4 = 0$

$x + 3 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x - 4) (x + 3) = 0$ true.

$x = 4, - 3$

Solve for x e^(x^2)=(e^x)*e^12

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