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

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

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

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

Rewrite $e^{2 x}$ as exponentiation.

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

Substitute $u$ for $e^{x}$ .

$u^{2} - u = 2$

Solve for $u$ .

Subtract $2$ from both sides of the equation.

$u^{2} - u - 2 = 0$

Factor $u^{2} - u - 2$ 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 $- 1$ .

$- 2, 1$

Write the factored form using these integers.

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

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

$u - 2 = 0$

$u + 1 = 0$

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

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

$u - 2 = 0$

Add $2$ to both sides of the equation.

$u = 2$

Set $u + 1$ equal to $0$ and solve for $u$ .

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Subtract $1$ from both sides of the equation.

$u = - 1$

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

$u = 2, - 1$

Substitute $2$ for $u$ in $u = e^{x}$ .

$2 = e^{x}$

Solve $2 = e^{x}$ .

Rewrite the equation as $e^{x} = 2$ .

$e^{x} = 2$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (2)$

Expand the left side.

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (2)$

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (2)$

Multiply $x$ by $1$ .

$x = \ln (2)$

Substitute $- 1$ for $u$ in $u = e^{x}$ .

$- 1 = e^{x}$

Solve $- 1 = e^{x}$ .

Rewrite the equation as $e^{x} = - 1$ .

$e^{x} = - 1$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (- 1)$

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

There is no solution for $e^{x} = - 1$

No solution

List the solutions that makes the equation true.

$x = \ln (2)$

The result can be shown in multiple forms.

Exact Form:

$x = \ln (2)$

Decimal Form:

$x = 0.69314718 \dots$

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

Related Questions

Support LCMGCF.com by disabling your adblocker.