Precalculus Examples

$6 e^{2 x} + 5 e^{x} - 4 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

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

$6 {(e^{x})}^{2} + 5 e^{x} - 4 = 0$

Step 1.2

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

$6 u^{2} + 5 u - 4 = 0$

Step 1.3

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 6 \cdot - 4 = - 24$ and whose sum is $b = 5$ .

Tap for more steps...

Step 1.3.1.1

Factor $5$ out of $5 u$ .

$6 u^{2} + 5 (u) - 4 = 0$

Step 1.3.1.2

Rewrite $5$ as $- 3$ plus $8$

$6 u^{2} + (- 3 + 8) u - 4 = 0$

Step 1.3.1.3

Apply the distributive property.

$6 u^{2} - 3 u + 8 u - 4 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.3.2.1

Group the first two terms and the last two terms.

$(6 u^{2} - 3 u) + 8 u - 4 = 0$

Step 1.3.2.2

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

$3 u (2 u - 1) + 4 (2 u - 1) = 0$

Step 1.3.3

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

$(2 u - 1) (3 u + 4) = 0$

Step 1.4

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

$(2 e^{x} - 1) (3 e^{x} + 4) = 0$

Step 2

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

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

$3 e^{x} + 4 = 0$

Step 3

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

Tap for more steps...

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 $x$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

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.

Tap for more steps...

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 $3 e^{x} + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $3 e^{x} + 4$ equal to $0$ .

$3 e^{x} + 4 = 0$

Step 4.2

Solve $3 e^{x} + 4 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Subtract $4$ from both sides of the equation.

$3 e^{x} = - 4$

Step 4.2.2

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

$\ln (3 e^{x}) = \ln (- 4)$

Step 4.2.3

The equation cannot be solved because $\ln (- 4)$ is undefined.

Undefined

Step 4.2.4

There is no solution for $3 e^{x} = - 4$

No solution

Step 5

The final solution is all the values that make $(2 e^{x} - 1) (3 e^{x} + 4) = 0$ 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:

$x = - 0.69314718 \dots$