What are the solutions to the equation 1/2 =3x-4x^3 ?

The solutions to the equation 1/2 =3x-4x^3 are x\approx 0.17364…,x\approx 0.76604…,x\approx-0.93969…

Find the zeros of 1/2 =3x-4x^3

The zeros of 1/2 =3x-4x^3 are x\approx 0.17364…,x\approx 0.76604…,x\approx-0.93969…

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\frac{1}{2} = 3 x - 4 x^{3}

x \approx 0.17364 \dots, x \approx 0.76604 \dots, x \approx - 0.93969 \dots

Solution steps

\frac{1}{2} = 3 x - 4 x^{3}

Multiply both sides by

2

1 = 6 x - 8 x^{3}

Switch sides

6 x - 8 x^{3} = 1

Move

1

to the left side

6 x - 8 x^{3} - 1 = 0

Write in the standard form

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

- 8 x^{3} + 6 x - 1 = 0

Find one solution for

- 8 x^{3} + 6 x - 1 = 0

using Newton-Raphson:

x \approx 0.17364 \dots

Apply long division:

\frac{- 8 x ^{3} + 6 x - 1}{x - 0.17364 \dots} = - 8 x^{2} - 1.38918 \dots x + 5.75877 \dots

- 8 x^{2} - 1.38918 \dots x + 5.75877 \dots \approx 0

Find one solution for

- 8 x^{2} - 1.38918 \dots x + 5.75877 \dots = 0

using Newton-Raphson:

x \approx 0.76604 \dots

Apply long division:

\frac{- 8 x ^{2} - 1.38918 \dots x + 5.75877 \dots}{x - 0.76604 \dots} = - 8 x - 7.51754 \dots

- 8 x - 7.51754 \dots \approx 0

x \approx - 0.93969 \dots

The solutions are

x \approx 0.17364 \dots, x \approx 0.76604 \dots, x \approx - 0.93969 \dots

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

(x^{2} + x)^{2} - 8 (x^{2} + x) + 12 = 0

725 = 5 x^{3}

27 x^{3} = 8

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

What are the solutions to the equation 1/2 =3x-4x^3 ?
The solutions to the equation 1/2 =3x-4x^3 are x\approx 0.17364…,x\approx 0.76604…,x\approx-0.93969…
Find the zeros of 1/2 =3x-4x^3
The zeros of 1/2 =3x-4x^3 are x\approx 0.17364…,x\approx 0.76604…,x\approx-0.93969…