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Popular >

x(1-x)^2= 1/10

Solution

x (1 - x)^{2} = \frac{1}{10}

Solution

x \approx 0.13304 \dots, x \approx 0.58739 \dots, x \approx 1.27955 \dots

Solution steps

x (1 - x)^{2} = \frac{1}{10}

Multiply both sides by

10

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

Expand

10 x (1 - x)^{2} : 10 x - 20 x^{2} + 10 x^{3}

10 x - 20 x^{2} + 10 x^{3} = 1

Move

1

to the left side

10 x - 20 x^{2} + 10 x^{3} - 1 = 0

Write in the standard form

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

10 x^{3} - 20 x^{2} + 10 x - 1 = 0

Find one solution for

10 x^{3} - 20 x^{2} + 10 x - 1 = 0

using Newton-Raphson:

x \approx 0.13304 \dots

Apply long division:

\frac{10 x ^{3} - 20 x ^{2} + 10 x - 1}{x - 0.13304 \dots} = 10 x^{2} - 18.66951 \dots x + 7.51604 \dots

10 x^{2} - 18.66951 \dots x + 7.51604 \dots \approx 0

Find one solution for

10 x^{2} - 18.66951 \dots x + 7.51604 \dots = 0

using Newton-Raphson:

x \approx 0.58739 \dots

Apply long division:

\frac{10 x ^{2} - 18.66951 \dots x + 7.51604 \dots}{x - 0.58739 \dots} = 10 x - 12.79556 \dots

10 x - 12.79556 \dots \approx 0

x \approx 1.27955 \dots

The solutions are

x \approx 0.13304 \dots, x \approx 0.58739 \dots, x \approx 1.27955 \dots

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