What are the solutions to the equation x^7=x+1 ?

The solutions to the equation x^7=x+1 are x\approx 1.11277…

Find the zeros of x^7=x+1

The zeros of x^7=x+1 are x\approx 1.11277…

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x^{7} = x + 1

x \approx 1.11277 \dots

Solution steps

x^{7} = x + 1

Move

1

to the left side

x^{7} - 1 = x

Move

x

to the left side

x^{7} - 1 - x = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

x^{7} - x - 1 = 0

Find one solution for

x^{7} - x - 1 = 0

using Newton-Raphson:

x \approx 1.11277 \dots

Apply long division:

\frac{x ^{7} - x - 1}{x - 1.11277 \dots} = x^{6} + 1.11277 \dots x^{5} + 1.23826 \dots x^{4} + 1.37791 \dots x^{3} + 1.53331 \dots x^{2} + 1.70623 \dots x + 0.89865 \dots

x^{6} + 1.11277 \dots x^{5} + 1.23826 \dots x^{4} + 1.37791 \dots x^{3} + 1.53331 \dots x^{2} + 1.70623 \dots x + 0.89865 \dots \approx 0

Find one solution for

x^{6} + 1.11277 \dots x^{5} + 1.23826 \dots x^{4} + 1.37791 \dots x^{3} + 1.53331 \dots x^{2} + 1.70623 \dots x + 0.89865 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 1.11277 \dots

34 x^{4} - 7 x^{2} - 3 x + 1 = 0

x^{3} - 4 x^{2} - 3 x + 18 = 0

3 (c^{2})^{2} + 10 c - 6 = 0

x^{3} - 14 x + 8 = 0

y^{4} - 9 = 0

What are the solutions to the equation x^7=x+1 ?
The solutions to the equation x^7=x+1 are x\approx 1.11277…
Find the zeros of x^7=x+1
The zeros of x^7=x+1 are x\approx 1.11277…