Solve for x e^(2x)-8e^x+15=0

$e^{2 x} - 8 e^{x} + 15 = 0$

Step 1

Factor the left side of the equation.

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

${(e^{x})}^{2} - 8 e^{x} + 15 = 0$

Let $u = e^{x}$ . Substitute $u$ for all occurrences of $e^{x}$ .

$u^{2} - 8 u + 15 = 0$

Factor $u^{2} - 8 u + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$(u - 5) (u - 3) = 0$

Replace all occurrences of $u$ with $e^{x}$ .

$(e^{x} - 5) (e^{x} - 3) = 0$

Step 2

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

$e^{x} - 5 = 0$

$e^{x} - 3 = 0$

Step 3

Set $e^{x} - 5$ equal to $0$ and solve for $x$ .

Set $e^{x} - 5$ equal to $0$ .

$e^{x} - 5 = 0$

Solve $e^{x} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$e^{x} = 5$

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

$\ln (e^{x}) = \ln (5)$

Expand the left side.

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

$x \ln (e) = \ln (5)$

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

$x \cdot 1 = \ln (5)$

Multiply $x$ by $1$ .

$x = \ln (5)$

Step 4

Set $e^{x} - 3$ equal to $0$ and solve for $x$ .

Set $e^{x} - 3$ equal to $0$ .

$e^{x} - 3 = 0$

Solve $e^{x} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$e^{x} = 3$

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

$\ln (e^{x}) = \ln (3)$

Expand the left side.

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

$x \ln (e) = \ln (3)$

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

$x \cdot 1 = \ln (3)$

Multiply $x$ by $1$ .

$x = \ln (3)$

Step 5

The final solution is all the values that make $(e^{x} - 5) (e^{x} - 3) = 0$ true.

$x = \ln (5), \ln (3)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (5), \ln (3)$

Decimal Form:

$x = 1.60943791 \dots, 1.09861228 \dots$

Solve for x e^(2x)-8e^x+15=0

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