What are the solutions to the equation x^5+2x^4+6x^2+1=114514 ?

The solutions to the equation x^5+2x^4+6x^2+1=114514 are x\approx 9.89308…

Find the zeros of x^5+2x^4+6x^2+1=114514

The zeros of x^5+2x^4+6x^2+1=114514 are x\approx 9.89308…

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x^{5} + 2 x^{4} + 6 x^{2} + 1 = 114514

x \approx 9.89308 \dots

Solution steps

x^{5} + 2 x^{4} + 6 x^{2} + 1 = 114514

Move

114514

to the left side

x^{5} + 2 x^{4} + 6 x^{2} - 114513 = 0

Find one solution for

x^{5} + 2 x^{4} + 6 x^{2} - 114513 = 0

using Newton-Raphson:

x \approx 9.89308 \dots

Apply long division:

\frac{x ^{5} + 2 x ^{4} + 6 x ^{2} - 114513}{x - 9.89308 \dots} = x^{4} + 11.89308 \dots x^{3} + 117.65934 \dots x^{2} + 1170.01417 \dots x + 11575.05219 \dots

x^{4} + 11.89308 \dots x^{3} + 117.65934 \dots x^{2} + 1170.01417 \dots x + 11575.05219 \dots \approx 0

Find one solution for

x^{4} + 11.89308 \dots x^{3} + 117.65934 \dots x^{2} + 1170.01417 \dots x + 11575.05219 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 9.89308 \dots

x^{3} \cdot 19.5 = 26

x^{3} - 8 = 19

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

z^{5} = - 163 + 16

x (x^{3} - 4 x^{2} + 6 x - 5) = 0

What are the solutions to the equation x^5+2x^4+6x^2+1=114514 ?
The solutions to the equation x^5+2x^4+6x^2+1=114514 are x\approx 9.89308…
Find the zeros of x^5+2x^4+6x^2+1=114514
The zeros of x^5+2x^4+6x^2+1=114514 are x\approx 9.89308…