Precalculus Examples

$2 e^{2 x} + 5 e^{x} - 3 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$2 {(e^{x})}^{2} + 5 e^{x} - 3 = 0$

Step 1.2

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

$2 u^{2} + 5 u - 3 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

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Step 1.3.1.1

Factor $5$ out of $5 u$ .

$2 u^{2} + 5 (u) - 3 = 0$

Step 1.3.1.2

Rewrite $5$ as $- 1$ plus $6$

$2 u^{2} + (- 1 + 6) u - 3 = 0$

Step 1.3.1.3

Apply the distributive property.

$2 u^{2} - 1 u + 6 u - 3 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(2 u^{2} - 1 u) + 6 u - 3 = 0$

Step 1.3.2.2

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

$u (2 u - 1) + 3 (2 u - 1) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(2 u - 1) (u + 3) = 0$

Step 1.4

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

$(2 e^{x} - 1) (e^{x} + 3) = 0$

Step 2

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

$2 e^{x} - 1 = 0$

$e^{x} + 3 = 0$

Step 3

Set $2 e^{x} - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 e^{x} - 1$ equal to $0$ .

$2 e^{x} - 1 = 0$

Step 3.2

Solve $2 e^{x} - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$2 e^{x} = 1$

Step 3.2.2

Divide each term in $2 e^{x} = 1$ by $2$ and simplify.

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Step 3.2.2.1

Divide each term in $2 e^{x} = 1$ by $2$ .

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $e^{x}$ by $1$ .

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

Step 3.2.3

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

$\ln (e^{x}) = \ln (\frac{1}{2})$

Step 3.2.4

Expand the left side.

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Step 3.2.4.1

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

$x \ln (e) = \ln (\frac{1}{2})$

Step 3.2.4.2

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

$x \cdot 1 = \ln (\frac{1}{2})$

Step 3.2.4.3

Multiply $x$ by $1$ .

$x = \ln (\frac{1}{2})$

Step 4

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

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Step 4.1

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

$e^{x} + 3 = 0$

Step 4.2

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

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$e^{x} = - 3$

Step 4.2.2

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

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

Step 4.2.3

The equation cannot be solved because $\ln (- 3)$ is undefined.

Undefined

Step 4.2.4

There is no solution for $e^{x} = - 3$

No solution

Step 5

The final solution is all the values that make $(2 e^{x} - 1) (e^{x} + 3) = 0$ true.

$x = \ln (\frac{1}{2})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\frac{1}{2})$

Decimal Form:

$x = - 0.69314718 \dots$