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

The solutions to the equation 3x^2(x-1)-2=x^2 are x\approx 1.59528…

Find the zeros of 3x^2(x-1)-2=x^2

The zeros of 3x^2(x-1)-2=x^2 are x\approx 1.59528…

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

x \approx 1.59528 \dots

Solution steps

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

Expand

3 x^{2} (x - 1) - 2 : 3 x^{3} - 3 x^{2} - 2

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

Move

x^{2}

to the left side

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

Find one solution for

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

using Newton-Raphson:

x \approx 1.59528 \dots

Apply long division:

\frac{3 x ^{3} - 4 x ^{2} - 2}{x - 1.59528 \dots} = 3 x^{2} + 0.78586 \dots x + 1.25369 \dots

3 x^{2} + 0.78586 \dots x + 1.25369 \dots \approx 0

Find one solution for

3 x^{2} + 0.78586 \dots x + 1.25369 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 1.59528 \dots

x^{4} + 4 x^{3} - 5 x^{2} - 16 x + 4 = 0

3 x^{5} + x^{4} - 9 x^{3} - 9 x^{2} - 2 x = 0

x + x^{3} = 4

x^{3} + 125 = 0

x^{4} + 13 x^{2} + 40 = 0

What are the solutions to the equation 3x^2(x-1)-2=x^2 ?
The solutions to the equation 3x^2(x-1)-2=x^2 are x\approx 1.59528…
Find the zeros of 3x^2(x-1)-2=x^2
The zeros of 3x^2(x-1)-2=x^2 are x\approx 1.59528…