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

The solutions to the equation (3x-1)^2x(x-3)=1 are x\approx-0.15059…,x\approx 3.00517…

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

The zeros of (3x-1)^2x(x-3)=1 are x\approx-0.15059…,x\approx 3.00517…

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

x \approx - 0.15059 \dots, x \approx 3.00517 \dots

Solution steps

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

Expand

(3 x - 1)^{2} x (x - 3) : 9 x^{4} - 33 x^{3} + 19 x^{2} - 3 x

9 x^{4} - 33 x^{3} + 19 x^{2} - 3 x = 1

Move

1

to the left side

9 x^{4} - 33 x^{3} + 19 x^{2} - 3 x - 1 = 0

Find one solution for

9 x^{4} - 33 x^{3} + 19 x^{2} - 3 x - 1 = 0

using Newton-Raphson:

x \approx - 0.15059 \dots

Apply long division:

\frac{9 x ^{4} - 33 x ^{3} + 19 x ^{2} - 3 x - 1}{x + 0.15059 \dots} = 9 x^{3} - 34.35534 \dots x^{2} + 24.17368 \dots x - 6.64039 \dots

9 x^{3} - 34.35534 \dots x^{2} + 24.17368 \dots x - 6.64039 \dots \approx 0

Find one solution for

9 x^{3} - 34.35534 \dots x^{2} + 24.17368 \dots x - 6.64039 \dots = 0

using Newton-Raphson:

x \approx 3.00517 \dots

Apply long division:

\frac{9 x ^{3} - 34.35534 \dots x ^{2} + 24.17368 \dots x - 6.64039 \dots}{x - 3.00517 \dots} = 9 x^{2} - 7.30872 \dots x + 2.20965 \dots

9 x^{2} - 7.30872 \dots x + 2.20965 \dots \approx 0

Find one solution for

9 x^{2} - 7.30872 \dots x + 2.20965 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx - 0.15059 \dots, x \approx 3.00517 \dots

z^{3} - 3 z - 3 = 0

- 2 (x + 2)^{3} - 8 = 0

x, f = x^{4} - 6 x^{2} + 6

x^{3} + 3 x^{2} + 7 x + 21 = 0

z^{3} + z + 1 = 0

What are the solutions to the equation (3x-1)^2x(x-3)=1 ?
The solutions to the equation (3x-1)^2x(x-3)=1 are x\approx-0.15059…,x\approx 3.00517…
Find the zeros of (3x-1)^2x(x-3)=1
The zeros of (3x-1)^2x(x-3)=1 are x\approx-0.15059…,x\approx 3.00517…