What is the answer to -e^{2x}+3x+4=0 ?

The answer to -e^{2x}+3x+4=0 is x=-(3\W_{-1}(-2/(3e^{8/3)})+8)/6 ,x=-(3\W_{0}(-2/(3e^{8/3)})+8)/6

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-e^{2x}+3x+4=0

- e^{2 x} + 3 x + 4 = 0

x = - \frac{3 W _{- 1} ( - \frac{2}{3 e ^{\frac{8}{3}}} ) + 8}{6}, x = - \frac{3 W _{0} ( - \frac{2}{3 e ^{\frac{8}{3}}} ) + 8}{6}

Decimal Notation

x = 0.96550 \dots, x = - 1.30901 \dots

Solution steps

- e^{2 x} + 3 x + 4 = 0

Prepare

- e^{2 x} + 3 x + 4 = 0

for Lambert form:

- 1 = - e^{- 2 x} (3 x + 4)

Rewrite the equation with

- 2 x - \frac{8}{3} = u

and

x = - \frac{3 u + 8}{6}

- 1 = - e^{- 2 (- \frac{3 u + 8}{6})} (3 (- \frac{3 u + 8}{6}) + 4)

Simplify

- 1 = - e^{\frac{3 u + 8}{3}} (- \frac{3 u + 8}{2} + 4)

Rewrite

- 1 = - e^{\frac{3 u + 8}{3}} (- \frac{3 u + 8}{2} + 4)

in Lambert form:

e^{u} u = - \frac{2}{3 e ^{\frac{8}{3}}}

Solve

e^{u} u = - \frac{2}{3 e ^{\frac{8}{3}}} : u = W_{- 1} (- \frac{2}{3 e ^{\frac{8}{3}}}), u = W_{0} (- \frac{2}{3 e ^{\frac{8}{3}}})

u = W_{- 1} (- \frac{2}{3 e ^{\frac{8}{3}}}), u = W_{0} (- \frac{2}{3 e ^{\frac{8}{3}}})

Substitute back

u = - 2 x - \frac{8}{3},

solve for

x

Solve

- 2 x - \frac{8}{3} = W_{- 1} (- \frac{2}{3 e ^{\frac{8}{3}}}) : x = - \frac{3 W _{- 1} ( - \frac{2}{3 e ^{\frac{8}{3}}} ) + 8}{6}

Solve

- 2 x - \frac{8}{3} = W_{0} (- \frac{2}{3 e ^{\frac{8}{3}}}) : x = - \frac{3 W _{0} ( - \frac{2}{3 e ^{\frac{8}{3}}} ) + 8}{6}

x = - \frac{3 W _{- 1} ( - \frac{2}{3 e ^{\frac{8}{3}}} ) + 8}{6}, x = - \frac{3 W _{0} ( - \frac{2}{3 e ^{\frac{8}{3}}} ) + 8}{6}

\frac{\frac{( \frac{2}{3} ) ^{n}}{4}}{9} = (\frac{2}{3})^{4}

\frac{( 3 ^{2 x + 3} )}{( 9 ^{3} )} = 9^{14}

b, 4^{l o g_{2} (a^{2} b)} = 3 a^{3}

3^{x} - 1 + 3^{x} - 2 + 3^{x} - 3 = 13

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

What is the answer to -e^{2x}+3x+4=0 ?
The answer to -e^{2x}+3x+4=0 is x=-(3\W_{-1}(-2/(3e^{8/3)})+8)/6 ,x=-(3\W_{0}(-2/(3e^{8/3)})+8)/6