Precalculus Examples

$e^{2 x} + 7 e^{x} - 3 = 0$

Step 1

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

${(e^{x})}^{2} + 7 e^{x} - 3 = 0$

Step 2

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

$u^{2} + 7 u - 3 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.2

Substitute the values $a = 1$ , $b = 7$ , and $c = - 3$ into the quadratic formula and solve for $u$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot - 3)}}{2 \cdot 1}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Raise $7$ to the power of $2$ .

$u = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 3.3.1.2

Multiply $- 4 \cdot 1 \cdot - 3$ .

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Step 3.3.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 3}}{2 \cdot 1}$

Step 3.3.1.2.2

Multiply $- 4$ by $- 3$ .

$u = \frac{- 7 \pm \sqrt{49 + 12}}{2 \cdot 1}$

Step 3.3.1.3

Add $49$ and $12$ .

$u = \frac{- 7 \pm \sqrt{61}}{2 \cdot 1}$

Step 3.3.2

Multiply $2$ by $1$ .

$u = \frac{- 7 \pm \sqrt{61}}{2}$

Step 3.4

The final answer is the combination of both solutions.

$u = - \frac{7 - \sqrt{61}}{2}, - \frac{7 + \sqrt{61}}{2}$

Step 4

Substitute $- \frac{7 - \sqrt{61}}{2}$ for $u$ in $u = e^{x}$ .

$- \frac{7 - \sqrt{61}}{2} = e^{x}$

Step 5

Solve $- \frac{7 - \sqrt{61}}{2} = e^{x}$ .

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Step 5.1

Rewrite the equation as $e^{x} = - \frac{7 - \sqrt{61}}{2}$ .

$e^{x} = - \frac{7 - \sqrt{61}}{2}$

Step 5.2

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

$\ln (e^{x}) = \ln (- \frac{7 - \sqrt{61}}{2})$

Step 5.3

Expand the left side.

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Step 5.3.1

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

$x \ln (e) = \ln (- \frac{7 - \sqrt{61}}{2})$

Step 5.3.2

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

$x \cdot 1 = \ln (- \frac{7 - \sqrt{61}}{2})$

Step 5.3.3

Multiply $x$ by $1$ .

$x = \ln (- \frac{7 - \sqrt{61}}{2})$

Step 6

Substitute $- \frac{7 + \sqrt{61}}{2}$ for $u$ in $u = e^{x}$ .

$- \frac{7 + \sqrt{61}}{2} = e^{x}$

Step 7

Solve $- \frac{7 + \sqrt{61}}{2} = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = - \frac{7 + \sqrt{61}}{2}$ .

$e^{x} = - \frac{7 + \sqrt{61}}{2}$

Step 7.2

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

$\ln (e^{x}) = \ln (- \frac{7 + \sqrt{61}}{2})$

Step 7.3

The equation cannot be solved because $\ln (- \frac{7 + \sqrt{61}}{2})$ is undefined.

Undefined

Step 7.4

There is no solution for $e^{x} = - \frac{7 + \sqrt{61}}{2}$

No solution

Step 8

List the solutions that makes the equation true.

$x = \ln (- \frac{7 - \sqrt{61}}{2})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \ln (- \frac{7 - \sqrt{61}}{2})$

Decimal Form:

$x = - 0.90356001 \dots$