Solve for x 6e^(6x)-17e^(3x)+7=0

$6 e^{6 x} - 17 e^{3 x} + 7 = 0$

Factor the left side of the equation.

Rewrite $e^{6 x}$ as ${(e^{3 x})}^{2}$ .

$6 {(e^{3 x})}^{2} - 17 e^{3 x} + 7 = 0$

Let $u = e^{3 x}$ . Substitute $u$ for all occurrences of $e^{3 x}$ .

$6 u^{2} - 17 u + 7 = 0$

Factor by grouping.

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 $a \cdot c = 6 \cdot 7 = 42$ and whose sum is $b = - 17$ .

Factor $- 17$ out of $- 17 u$ .

$6 u^{2} - 17 u + 7 = 0$

Rewrite $- 17$ as $- 3$ plus $- 14$

$6 u^{2} + (- 3 - 14) u + 7 = 0$

Apply the distributive property.

$6 u^{2} - 3 u - 14 u + 7 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 u^{2} - 3 u) - 14 u + 7 = 0$

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

$3 u (2 u - 1) - 7 (2 u - 1) = 0$

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

$(2 u - 1) (3 u - 7) = 0$

Replace all occurrences of $u$ with $e^{3 x}$ .

$(2 e^{3 x} - 1) (3 e^{3 x} - 7) = 0$

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

$2 e^{3 x} - 1 = 0$

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

Set $2 e^{3 x} - 1$ equal to $0$ and solve for $x$ .

Set $2 e^{3 x} - 1$ equal to $0$ .

$2 e^{3 x} - 1 = 0$

Solve $2 e^{3 x} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 e^{3 x} = 1$

Divide each term in $2 e^{3 x} = 1$ by $2$ and simplify.

Divide each term in $2 e^{3 x} = 1$ by $2$ .

$\frac{2 e^{3 x}}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 e^{3 x}}{2} = \frac{1}{2}$

Divide $e^{3 x}$ by $1$ .

$e^{3 x} = \frac{1}{2}$

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

$\ln (e^{3 x}) = \ln (\frac{1}{2})$

Expand the left side.

Expand $\ln (e^{3 x})$ by moving $3 x$ outside the logarithm.

$3 x \ln (e) = \ln (\frac{1}{2})$

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

$3 x \cdot 1 = \ln (\frac{1}{2})$

Multiply $3$ by $1$ .

$3 x = \ln (\frac{1}{2})$

Divide each term in $3 x = \ln (\frac{1}{2})$ by $3$ and simplify.

Divide each term in $3 x = \ln (\frac{1}{2})$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (\frac{1}{2})}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\ln (\frac{1}{2})}{3}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{2})}{3}$

Set $3 e^{3 x} - 7$ equal to $0$ and solve for $x$ .

Set $3 e^{3 x} - 7$ equal to $0$ .

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

Solve $3 e^{3 x} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$3 e^{3 x} = 7$

Divide each term in $3 e^{3 x} = 7$ by $3$ and simplify.

Divide each term in $3 e^{3 x} = 7$ by $3$ .

$\frac{3 e^{3 x}}{3} = \frac{7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 e^{3 x}}{3} = \frac{7}{3}$

Divide $e^{3 x}$ by $1$ .

$e^{3 x} = \frac{7}{3}$

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

$\ln (e^{3 x}) = \ln (\frac{7}{3})$

Expand the left side.

Expand $\ln (e^{3 x})$ by moving $3 x$ outside the logarithm.

$3 x \ln (e) = \ln (\frac{7}{3})$

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

$3 x \cdot 1 = \ln (\frac{7}{3})$

Multiply $3$ by $1$ .

$3 x = \ln (\frac{7}{3})$

Divide each term in $3 x = \ln (\frac{7}{3})$ by $3$ and simplify.

Divide each term in $3 x = \ln (\frac{7}{3})$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (\frac{7}{3})}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\ln (\frac{7}{3})}{3}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{7}{3})}{3}$

The final solution is all the values that make $(2 e^{3 x} - 1) (3 e^{3 x} - 7) = 0$ true.

$x = \frac{\ln (\frac{1}{2})}{3}, \frac{\ln (\frac{7}{3})}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{1}{2})}{3}, \frac{\ln (\frac{7}{3})}{3}$

Decimal Form:

$x = - 0.23104906 \dots, 0.28243262 \dots$

Solve for x 6e^(6x)-17e^(3x)+7=0

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