What are the solutions to the equation x^2+1=4x^4x ?

The solutions to the equation x^2+1=4x^4x are x\approx 0.84391…

Find the zeros of x^2+1=4x^4x

The zeros of x^2+1=4x^4x are x\approx 0.84391…

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x^{2} + 1 = 4 x^{4} x

x \approx 0.84391 \dots

Solution steps

x^{2} + 1 = 4 x^{4} x

Expand

4 x^{4} x : 4 x^{5}

x^{2} + 1 = 4 x^{5}

Switch sides

4 x^{5} = x^{2} + 1

Move

1

to the left side

4 x^{5} - 1 = x^{2}

Move

x^{2}

to the left side

4 x^{5} - 1 - x^{2} = 0

Write in the standard form

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

4 x^{5} - x^{2} - 1 = 0

Find one solution for

4 x^{5} - x^{2} - 1 = 0

using Newton-Raphson:

x \approx 0.84391 \dots

Apply long division:

\frac{4 x ^{5} - x ^{2} - 1}{x - 0.84391 \dots} = 4 x^{4} + 3.37565 \dots x^{3} + 2.84876 \dots x^{2} + 1.40411 \dots x + 1.18495 \dots

4 x^{4} + 3.37565 \dots x^{3} + 2.84876 \dots x^{2} + 1.40411 \dots x + 1.18495 \dots \approx 0

Find one solution for

4 x^{4} + 3.37565 \dots x^{3} + 2.84876 \dots x^{2} + 1.40411 \dots x + 1.18495 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 0.84391 \dots

2 - x = x^{4}

4 x^{4} + x^{3} + x^{2} + x + 1 = 0

2 x^{3} + 3 x^{2} - 32 x + 15 = 0

3 x^{3} - 2 x^{2} - 82 x - 10 = 0

2 x^{4} - 12 x^{2} - 54 = 0

What are the solutions to the equation x^2+1=4x^4x ?
The solutions to the equation x^2+1=4x^4x are x\approx 0.84391…
Find the zeros of x^2+1=4x^4x
The zeros of x^2+1=4x^4x are x\approx 0.84391…