Trigonometry Examples

$e^{2 x} - 30 e^{x} + 1 = 0$

Step 1

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

${(e^{x})}^{2} - 30 e^{x} + 1 = 0$

Step 2

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

$u^{2} - 30 u + 1 = 0$

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 $a = 1$ , $b = - 30$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

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 $- 30$ to the power of $2$ .

$u = \frac{30 \pm \sqrt{900 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{30 \pm \sqrt{900 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.3.1.2.2

Multiply $- 4$ by $1$ .

$u = \frac{30 \pm \sqrt{900 - 4}}{2 \cdot 1}$

Step 3.3.1.3

Subtract $4$ from $900$ .

$u = \frac{30 \pm \sqrt{896}}{2 \cdot 1}$

Step 3.3.1.4

Rewrite $896$ as $8^{2} \cdot 14$ .

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Step 3.3.1.4.1

Factor $64$ out of $896$ .

$u = \frac{30 \pm \sqrt{64 (14)}}{2 \cdot 1}$

Step 3.3.1.4.2

Rewrite $64$ as $8^{2}$ .

$u = \frac{30 \pm \sqrt{8^{2} \cdot 14}}{2 \cdot 1}$

Step 3.3.1.5

Pull terms out from under the radical.

$u = \frac{30 \pm 8 \sqrt{14}}{2 \cdot 1}$

Step 3.3.2

Multiply $2$ by $1$ .

$u = \frac{30 \pm 8 \sqrt{14}}{2}$

Step 3.3.3

Simplify $\frac{30 \pm 8 \sqrt{14}}{2}$ .

$u = 15 \pm 4 \sqrt{14}$

Step 3.4

The final answer is the combination of both solutions.

$u = 15 + 4 \sqrt{14}, 15 - 4 \sqrt{14}$

Step 4

Substitute $15 + 4 \sqrt{14}$ for $u$ in $u = e^{x}$ .

$15 + 4 \sqrt{14} = e^{x}$

Step 5

Solve $15 + 4 \sqrt{14} = e^{x}$ .

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Step 5.1

Rewrite the equation as $e^{x} = 15 + 4 \sqrt{14}$ .

$e^{x} = 15 + 4 \sqrt{14}$

Step 5.2

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

$\ln (e^{x}) = \ln (15 + 4 \sqrt{14})$

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.

$x \ln (e) = \ln (15 + 4 \sqrt{14})$

Step 5.3.2

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

$x \cdot 1 = \ln (15 + 4 \sqrt{14})$

Step 5.3.3

Multiply $x$ by $1$ .

$x = \ln (15 + 4 \sqrt{14})$

Step 6

Substitute $15 - 4 \sqrt{14}$ for $u$ in $u = e^{x}$ .

$15 - 4 \sqrt{14} = e^{x}$

Step 7

Solve $15 - 4 \sqrt{14} = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = 15 - 4 \sqrt{14}$ .

$e^{x} = 15 - 4 \sqrt{14}$

Step 7.2

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

$\ln (e^{x}) = \ln (15 - 4 \sqrt{14})$

Step 7.3

Expand the left side.

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Step 7.3.1

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

$x \ln (e) = \ln (15 - 4 \sqrt{14})$

Step 7.3.2

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

$x \cdot 1 = \ln (15 - 4 \sqrt{14})$

Step 7.3.3

Multiply $x$ by $1$ .

$x = \ln (15 - 4 \sqrt{14})$

Step 8

List the solutions that makes the equation true.

$x = \ln (15 + 4 \sqrt{14}), \ln (15 - 4 \sqrt{14})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \ln (15 + 4 \sqrt{14}), \ln (15 - 4 \sqrt{14})$

Decimal Form:

$x = 3.40008441 \dots, - 3.40008441 \dots$