What are the solutions to the equation x^3=1+x+x^2 ?

The solutions to the equation x^3=1+x+x^2 are x\approx 1.83928…

Find the zeros of x^3=1+x+x^2

The zeros of x^3=1+x+x^2 are x\approx 1.83928…

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

x \approx 1.83928 \dots

Solution steps

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

Move

x^{2}

to the left side

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

Move

x

to the left side

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

Move

1

to the left side

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

Find one solution for

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

using Newton-Raphson:

x \approx 1.83928 \dots

Apply long division:

\frac{x ^{3} - x ^{2} - x - 1}{x - 1.83928 \dots} = x^{2} + 0.83928 \dots x + 0.54368 \dots

x^{2} + 0.83928 \dots x + 0.54368 \dots \approx 0

Find one solution for

x^{2} + 0.83928 \dots x + 0.54368 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 1.83928 \dots

2 x^{3} - 2 x^{2} - 24 x = 0

y^{2} y^{2} + y + 32 = (y - 4)^{2}

3 x^{3} - 5 x^{2} + 6 x = 0

- x^{3} - 5 x^{2} + 6 x + 32 = 0

x^{3} - x - 12 = 0

What are the solutions to the equation x^3=1+x+x^2 ?
The solutions to the equation x^3=1+x+x^2 are x\approx 1.83928…
Find the zeros of x^3=1+x+x^2
The zeros of x^3=1+x+x^2 are x\approx 1.83928…