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Popular >

e^x+x+1=0

Solution

e^{x} + x + 1 = 0

Solution

x = - W_{0} (\frac{1}{e}) - 1

+1

Decimal Notation

x = - 1.27846 \dots

Solution steps

e^{x} + x + 1 = 0

Prepare

e^{x} + x + 1 = 0

for Lambert form:

1 = - e^{- x} (x + 1)

Rewrite the equation with

- x - 1 = u

and

x = - u - 1

1 = - e^{- (- u - 1)} ((- u - 1) + 1)

Simplify

1 = e^{u + 1} u

Rewrite

1 = e^{u + 1} u

in Lambert form:

e^{u} u = \frac{1}{e}

Solve

e^{u} u = \frac{1}{e} : u = W_{0} (\frac{1}{e})

u = W_{0} (\frac{1}{e})

Substitute back

u = - x - 1,

solve for

x

Solve

- x - 1 = W_{0} (\frac{1}{e}) : x = - W_{0} (\frac{1}{e}) - 1

x = - W_{0} (\frac{1}{e}) - 1

Popular Examples

12=-82e^{-4k}+90

12 = - 82 e^{- 4 k} + 90

xe^{x-6}=e^{-5}

x e^{x - 6} = e^{- 5}

5^{- 2 x} = 3

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

8=60t-1800+(1800)/(e^{t/(30))}

8 = 60 t - 1800 + \frac{1800}{e ^{\frac{t}{30}}}

Frequently Asked Questions (FAQ)

What is the answer to e^x+x+1=0 ?
The answer to e^x+x+1=0 is x=-\W_{0}(1/e)-1