Algebra Examples

$3 e^{4 x} - 9 e^{2 x} - 15 = 0$

Step 1

Rewrite $e^{4 x}$ as exponentiation.

$3 {(e^{x})}^{4} - 9 e^{2 x} - 15 = 0$

Step 2

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

$3 {(e^{x})}^{4} - 9 {(e^{x})}^{2} - 15 = 0$

Step 3

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

$3 u^{4} - 9 u^{2} - 15 = 0$

Step 4

Solve for $u$ .

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Step 4.1

Substitute $u = u^{2}$ into the equation. This will make the quadratic formula easy to use.

$3 u^{2} - 9 u - 15 = 0$

$u = u^{2}$

Step 4.2

Factor $3$ out of $3 u^{2} - 9 u - 15$ .

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Step 4.2.1

Factor $3$ out of $3 u^{2}$ .

$3 (u^{2}) - 9 u - 15 = 0$

Step 4.2.2

Factor $3$ out of $- 9 u$ .

$3 (u^{2}) + 3 (- 3 u) - 15 = 0$

Step 4.2.3

Factor $3$ out of $- 15$ .

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

Step 4.2.4

Factor $3$ out of $3 u^{2} + 3 (- 3 u)$ .

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

Step 4.2.5

Factor $3$ out of $3 (u^{2} - 3 u) + 3 \cdot - 5$ .

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

Step 4.3

Divide each term in $3 (u^{2} - 3 u - 5) = 0$ by $3$ and simplify.

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Step 4.3.1

Divide each term in $3 (u^{2} - 3 u - 5) = 0$ by $3$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $u^{2} - 3 u - 5$ by $1$ .

$u^{2} - 3 u - 5 = \frac{0}{3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $0$ by $3$ .

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

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 5$ into the quadratic formula and solve for $u$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 3$ to the power of $2$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 1}$

Step 4.6.1.2.2

Multiply $- 4$ by $- 5$ .

$u = \frac{3 \pm \sqrt{9 + 20}}{2 \cdot 1}$

Step 4.6.1.3

Add $9$ and $20$ .

$u = \frac{3 \pm \sqrt{29}}{2 \cdot 1}$

Step 4.6.2

Multiply $2$ by $1$ .

$u = \frac{3 \pm \sqrt{29}}{2}$

Step 4.7

The final answer is the combination of both solutions.

$u = \frac{3 + \sqrt{29}}{2}, \frac{3 - \sqrt{29}}{2}$

Step 4.8

Substitute the real value of $u = u^{2}$ back into the solved equation.

$u^{2} = 4.1925824$

${(u^{2})}^{1} = - 1.1925824$

Step 4.9

Solve the first equation for $u$ .

$u^{2} = 4.1925824$

Step 4.10

Solve the equation for $u$ .

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Step 4.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{4.1925824}$

Step 4.10.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.10.2.1

First, use the positive value of the $\pm$ to find the first solution.

$u = \sqrt{4.1925824}$

Step 4.10.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - \sqrt{4.1925824}$

Step 4.10.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = \sqrt{4.1925824}, - \sqrt{4.1925824}$

Step 4.11

Solve the second equation for $u$ .

${(u^{2})}^{1} = - 1.1925824$

Step 4.12

Solve the equation for $u$ .

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Step 4.12.1

Remove parentheses.

$u^{2} = - 1.1925824$

Step 4.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{- 1.1925824}$

Step 4.12.3

Simplify $\pm \sqrt{- 1.1925824}$ .

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Step 4.12.3.1

Rewrite $- 1.1925824$ as $- 1 (1.1925824)$ .

$u = \pm \sqrt{- 1 (1.1925824)}$

Step 4.12.3.2

Rewrite $\sqrt{- 1 (1.1925824)}$ as $\sqrt{- 1} \cdot \sqrt{1.1925824}$ .

$u = \pm \sqrt{- 1} \cdot \sqrt{1.1925824}$

Step 4.12.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \pm i \sqrt{1.1925824}$

Step 4.12.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.12.4.1

First, use the positive value of the $\pm$ to find the first solution.

$u = i \sqrt{1.1925824}$

Step 4.12.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - i \sqrt{1.1925824}$

Step 4.12.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = i \sqrt{1.1925824}, - i \sqrt{1.1925824}$

Step 4.13

The solution to $3 u^{4} - 9 u^{2} - 15 = 0$ is $u = \sqrt{4.1925824}, - \sqrt{4.1925824}, i \sqrt{1.1925824}, - i \sqrt{1.1925824}$ .

$u = \sqrt{4.1925824}, - \sqrt{4.1925824}, i \sqrt{1.1925824}, - i \sqrt{1.1925824}$

Step 5

Substitute $\sqrt{4.1925824}$ for $u$ in $u = e^{x}$ .

$\sqrt{4.1925824} = e^{x}$

Step 6

Solve $\sqrt{4.1925824} = e^{x}$ .

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Step 6.1

Rewrite the equation as $e^{x} = \sqrt{4.1925824}$ .

$e^{x} = \sqrt{4.1925824}$

Step 6.2

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

$\ln (e^{x}) = \ln (\sqrt{4.1925824})$

Step 6.3

Expand the left side.

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Step 6.3.1

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

$x \ln (e) = \ln (\sqrt{4.1925824})$

Step 6.3.2

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

$x \cdot 1 = \ln (\sqrt{4.1925824})$

Step 6.3.3

Multiply $x$ by $1$ .

$x = \ln (\sqrt{4.1925824})$

Step 7

Substitute $- \sqrt{4.1925824}$ for $u$ in $u = e^{x}$ .

$- \sqrt{4.1925824} = e^{x}$

Step 8

Solve $- \sqrt{4.1925824} = e^{x}$ .

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Step 8.1

Rewrite the equation as $e^{x} = - \sqrt{4.1925824}$ .

$e^{x} = - \sqrt{4.1925824}$

Step 8.2

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

$\ln (e^{x}) = \ln (- \sqrt{4.1925824})$

Step 8.3

The equation cannot be solved because $\ln (- \sqrt{4.1925824})$ is undefined.

Undefined

Step 8.4

There is no solution for $e^{x} = - \sqrt{4.1925824}$

No solution

Step 9

Substitute $i \sqrt{1.1925824}$ for $u$ in $u = e^{x}$ .

$i \sqrt{1.1925824} = e^{x}$

Step 10

Solve $i \sqrt{1.1925824} = e^{x}$ .

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Step 10.1

Rewrite the equation as $e^{x} = i \sqrt{1.1925824}$ .

$e^{x} = i \sqrt{1.1925824}$

Step 10.2

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

$\ln (e^{x}) = \ln (i \sqrt{1.1925824})$

Step 10.3

Expand the left side.

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Step 10.3.1

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

$x \ln (e) = \ln (i \sqrt{1.1925824})$

Step 10.3.2

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

$x \cdot 1 = \ln (i \sqrt{1.1925824})$

Step 10.3.3

Multiply $x$ by $1$ .

$x = \ln (i \sqrt{1.1925824})$

Step 10.4

Expand the right side.

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Step 10.4.1

Rewrite $\ln (i \sqrt{1.1925824})$ as $\ln (i) + \ln (\sqrt{1.1925824})$ .

$x = \ln (i) + \ln (\sqrt{1.1925824})$

Step 10.4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1.1925824}$ as ${1.1925824}^{\frac{1}{2}}$ .

$x = \ln (i) + \ln ({1.1925824}^{\frac{1}{2}})$

Step 10.4.3

Expand $\ln ({1.1925824}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$x = \ln (i) + \frac{1}{2} \ln (1.1925824)$

Step 10.4.4

Combine $\frac{1}{2}$ and $\ln (1.1925824)$ .

$x = \ln (i) + \frac{\ln (1.1925824)}{2}$

Step 10.5

Simplify.

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Step 10.5.1

Simplify each term.

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Step 10.5.1.1

Rewrite $\frac{\ln (1.1925824)}{2}$ as $\frac{1}{2} \ln (1.1925824)$ .

$x = \ln (i) + \frac{1}{2} \ln (1.1925824)$

Step 10.5.1.2

Simplify $\frac{1}{2} \ln (1.1925824)$ by moving $\frac{1}{2}$ inside the logarithm.

$x = \ln (i) + \ln ({1.1925824}^{\frac{1}{2}})$

Step 10.5.1.3

Rewrite $1.1925824$ as ${1.09205421}^{2}$ .

$x = \ln (i) + \ln ({({1.09205421}^{2})}^{\frac{1}{2}})$

Step 10.5.1.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \ln (i) + \ln ({1.09205421}^{2 (\frac{1}{2})})$

Step 10.5.1.5

Cancel the common factor of $2$ .

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Step 10.5.1.5.1

Cancel the common factor.

$x = \ln (i) + \ln ({1.09205421}^{2 (\frac{1}{2})})$

Step 10.5.1.5.2

Rewrite the expression.

$x = \ln (i) + \ln ({1.09205421}^{1})$

Step 10.5.1.6

Evaluate the exponent.

$x = \ln (i) + \ln (1.09205421)$

Step 10.5.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$x = \ln (i \cdot 1.09205421)$

Step 10.5.3

Move $1.09205421$ to the left of $i$ .

$x = \ln (1.09205421 i)$

Step 11

Substitute $- i \sqrt{1.1925824}$ for $u$ in $u = e^{x}$ .

$- i \sqrt{1.1925824} = e^{x}$

Step 12

Solve $- i \sqrt{1.1925824} = e^{x}$ .

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Step 12.1

Rewrite the equation as $e^{x} = - i \sqrt{1.1925824}$ .

$e^{x} = - i \sqrt{1.1925824}$

Step 12.2

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

$\ln (e^{x}) = \ln (- i \sqrt{1.1925824})$

Step 12.3

The equation cannot be solved because $\ln (- i \sqrt{1.1925824})$ is undefined.

Undefined

Step 12.4

There is no solution for $e^{x} = - i \sqrt{1.1925824}$

No solution

Step 13

List the solutions that makes the equation true.

$x = \ln (\sqrt{4.1925824}), \ln (1.09205421 i)$