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

The solutions to the equation x=x^3-2x^2+1 are x\approx 0.55495…,x\approx-0.80193…,x\approx 2.24697…

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

The zeros of x=x^3-2x^2+1 are x\approx 0.55495…,x\approx-0.80193…,x\approx 2.24697…

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

x \approx 0.55495 \dots, x \approx - 0.80193 \dots, x \approx 2.24697 \dots

Solution steps

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

Switch sides

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

Move

x

to the left side

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

Write in the standard form

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

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

Find one solution for

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

using Newton-Raphson:

x \approx 0.55495 \dots

Apply long division:

\frac{x ^{3} - 2 x ^{2} - x + 1}{x - 0.55495 \dots} = x^{2} - 1.44504 \dots x - 1.80193 \dots

x^{2} - 1.44504 \dots x - 1.80193 \dots \approx 0

Find one solution for

x^{2} - 1.44504 \dots x - 1.80193 \dots = 0

using Newton-Raphson:

x \approx - 0.80193 \dots

Apply long division:

\frac{x ^{2} - 1.44504 \dots x - 1.80193 \dots}{x + 0.80193 \dots} = x - 2.24697 \dots

x - 2.24697 \dots \approx 0

x \approx 2.24697 \dots

The solutions are

x \approx 0.55495 \dots, x \approx - 0.80193 \dots, x \approx 2.24697 \dots

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

30 x^{4} - 30 x^{2} = 0

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

3 x^{4} - 7 x - 3 x^{2} + 5 x + 2 = 0

x^{4} - 2 x^{3} - 6 x^{2} = 0

What are the solutions to the equation x=x^3-2x^2+1 ?
The solutions to the equation x=x^3-2x^2+1 are x\approx 0.55495…,x\approx-0.80193…,x\approx 2.24697…
Find the zeros of x=x^3-2x^2+1
The zeros of x=x^3-2x^2+1 are x\approx 0.55495…,x\approx-0.80193…,x\approx 2.24697…