What are the solutions to the equation x+8x^4=2x^2-1/6 ?

The solutions to the equation x+8x^4=2x^2-1/6 are x\approx-0.13354…,x\approx-0.62930…

Find the zeros of x+8x^4=2x^2-1/6

The zeros of x+8x^4=2x^2-1/6 are x\approx-0.13354…,x\approx-0.62930…

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x + 8 x^{4} = 2 x^{2} - \frac{1}{6}

x \approx - 0.13354 \dots, x \approx - 0.62930 \dots

Solution steps

x + 8 x^{4} = 2 x^{2} - \frac{1}{6}

Multiply both sides by

6

6 x + 48 x^{4} = 12 x^{2} - 1

Move

1

to the left side

6 x + 48 x^{4} + 1 = 12 x^{2}

Move

12 x^{2}

to the left side

6 x + 48 x^{4} + 1 - 12 x^{2} = 0

Write in the standard form

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

48 x^{4} - 12 x^{2} + 6 x + 1 = 0

Find one solution for

48 x^{4} - 12 x^{2} + 6 x + 1 = 0

using Newton-Raphson:

x \approx - 0.13354 \dots

Apply long division:

\frac{48 x ^{4} - 12 x ^{2} + 6 x + 1}{x + 0.13354 \dots} = 48 x^{3} - 6.41008 \dots x^{2} - 11.14397 \dots x + 7.48820 \dots

48 x^{3} - 6.41008 \dots x^{2} - 11.14397 \dots x + 7.48820 \dots \approx 0

Find one solution for

48 x^{3} - 6.41008 \dots x^{2} - 11.14397 \dots x + 7.48820 \dots = 0

using Newton-Raphson:

x \approx - 0.62930 \dots

Apply long division:

\frac{48 x ^{3} - 6.41008 \dots x ^{2} - 11.14397 \dots x + 7.48820 \dots}{x + 0.62930 \dots} = 48 x^{2} - 36.61675 \dots x + 11.89915 \dots

48 x^{2} - 36.61675 \dots x + 11.89915 \dots \approx 0

Find one solution for

48 x^{2} - 36.61675 \dots x + 11.89915 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx - 0.13354 \dots, x \approx - 0.62930 \dots

x, - f = - x^{3}

- 24 x - 24 = x^{3} + 8 x^{2}

4 x (5 x^{2} - 3) = 0

20 x = x^{2} (x) + 36 x - 2 x (x) (x)

y, y^{2} - y^{4} = x

What are the solutions to the equation x+8x^4=2x^2-1/6 ?
The solutions to the equation x+8x^4=2x^2-1/6 are x\approx-0.13354…,x\approx-0.62930…
Find the zeros of x+8x^4=2x^2-1/6
The zeros of x+8x^4=2x^2-1/6 are x\approx-0.13354…,x\approx-0.62930…