Precalculus Examples

$8 e^{2 x} + 8 e^{x} = 6$

Step 1

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

$8 {(e^{x})}^{2} + 8 e^{x} = 6$

Step 2

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

$8 u^{2} + 8 u = 6$

Step 3

Solve for $u$ .

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Step 3.1

Subtract $6$ from both sides of the equation.

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

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $2$ out of $8 u^{2} + 8 u - 6$ .

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Step 3.2.1.1

Factor $2$ out of $8 u^{2}$ .

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

Step 3.2.1.2

Factor $2$ out of $8 u$ .

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

Step 3.2.1.3

Factor $2$ out of $- 6$ .

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

Step 3.2.1.4

Factor $2$ out of $2 (4 u^{2}) + 2 (4 u)$ .

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

Step 3.2.1.5

Factor $2$ out of $2 (4 u^{2} + 4 u) + 2 (- 3)$ .

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

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor by grouping.

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Step 3.2.2.1.1

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = 4$ .

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Step 3.2.2.1.1.1

Factor $4$ out of $4 u$ .

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

Step 3.2.2.1.1.2

Rewrite $4$ as $- 2$ plus $6$

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

Step 3.2.2.1.1.3

Apply the distributive property.

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

Step 3.2.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2.1.2.2

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

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

Step 3.2.2.1.3

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

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

Step 3.2.2.2

Remove unnecessary parentheses.

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

Step 3.3

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

$2 u - 1 = 0$

$2 u + 3 = 0$

Step 3.4

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 3.4.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 3.4.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 3.4.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 3.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 3.5

Set $2 u + 3$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $2 u + 3$ equal to $0$ .

$2 u + 3 = 0$

Step 3.5.2

Solve $2 u + 3 = 0$ for $u$ .

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Step 3.5.2.1

Subtract $3$ from both sides of the equation.

$2 u = - 3$

Step 3.5.2.2

Divide each term in $2 u = - 3$ by $2$ and simplify.

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Step 3.5.2.2.1

Divide each term in $2 u = - 3$ by $2$ .

$\frac{2 u}{2} = \frac{- 3}{2}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 3}{2}$

Step 3.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 3}{2}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{3}{2}$

Step 3.6

The final solution is all the values that make $2 (2 u - 1) (2 u + 3) = 0$ true.

$u = \frac{1}{2}, - \frac{3}{2}$

Step 4

Substitute $\frac{1}{2}$ for $u$ in $u = e^{x}$ .

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

Step 5

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

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Step 5.1

Rewrite the equation as $e^{x} = \frac{1}{2}$ .

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

Step 5.2

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

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

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.

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

Step 5.3.2

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

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

Step 5.3.3

Multiply $x$ by $1$ .

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

Step 6

Substitute $- \frac{3}{2}$ for $u$ in $u = e^{x}$ .

$- \frac{3}{2} = e^{x}$

Step 7

Solve $- \frac{3}{2} = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = - \frac{3}{2}$ .

$e^{x} = - \frac{3}{2}$

Step 7.2

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

$\ln (e^{x}) = \ln (- \frac{3}{2})$

Step 7.3

The equation cannot be solved because $\ln (- \frac{3}{2})$ is undefined.

Undefined

Step 7.4

There is no solution for $e^{x} = - \frac{3}{2}$

No solution

Step 8

List the solutions that makes the equation true.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.69314718 \dots$