What are the solutions to the equation x+x^2-x^4+x^8=125 ?

The solutions to the equation x+x^2-x^4+x^8=125 are x\approx 1.83973…,x\approx-1.84635…

Find the zeros of x+x^2-x^4+x^8=125

The zeros of x+x^2-x^4+x^8=125 are x\approx 1.83973…,x\approx-1.84635…

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x + x^{2} - x^{4} + x^{8} = 125

x \approx 1.83973 \dots, x \approx - 1.84635 \dots

Solution steps

x + x^{2} - x^{4} + x^{8} = 125

Move

125

to the left side

x + x^{2} - x^{4} + x^{8} - 125 = 0

Write in the standard form

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

x^{8} - x^{4} + x^{2} + x - 125 = 0

Find one solution for

x^{8} - x^{4} + x^{2} + x - 125 = 0

using Newton-Raphson:

x \approx 1.83973 \dots

Apply long division:

\frac{x ^{8} - x ^{4} + x ^{2} + x - 125}{x - 1.83973 \dots} = x^{7} + 1.83973 \dots x^{6} + 3.38461 \dots x^{5} + 6.22678 \dots x^{4} + 10.45562 \dots x^{3} + 19.23554 \dots x^{2} + 36.38825 \dots x + 67.94466 \dots

x^{7} + 1.83973 \dots x^{6} + 3.38461 \dots x^{5} + 6.22678 \dots x^{4} + 10.45562 \dots x^{3} + 19.23554 \dots x^{2} + 36.38825 \dots x + 67.94466 \dots \approx 0

Find one solution for

x^{7} + 1.83973 \dots x^{6} + 3.38461 \dots x^{5} + 6.22678 \dots x^{4} + 10.45562 \dots x^{3} + 19.23554 \dots x^{2} + 36.38825 \dots x + 67.94466 \dots = 0

using Newton-Raphson:

x \approx - 1.84635 \dots

Apply long division:

\frac{x ^{7} + 1.83973 \dots x ^{6} + 3.38461 \dots x ^{5} + 6.22678 \dots x ^{4} + 10.45562 \dots x ^{3} + 19.23554 \dots x ^{2} + 36.38825 \dots x + 67.94466 \dots}{x + 1.84635 \dots} = x^{6} - 0.00662 \dots x^{5} + 3.39684 \dots x^{4} - 0.04499 \dots x^{3} + 10.53870 \dots x^{2} - 0.22264 \dots x + 36.79933 \dots

x^{6} - 0.00662 \dots x^{5} + 3.39684 \dots x^{4} - 0.04499 \dots x^{3} + 10.53870 \dots x^{2} - 0.22264 \dots x + 36.79933 \dots \approx 0

Find one solution for

x^{6} - 0.00662 \dots x^{5} + 3.39684 \dots x^{4} - 0.04499 \dots x^{3} + 10.53870 \dots x^{2} - 0.22264 \dots x + 36.79933 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 1.83973 \dots, x \approx - 1.84635 \dots

s^{3} + 6 s^{2} + 5 s + 30 = 0

49 x^{4} + 9 = 58 x^{2}

z^{8} = 8 1^{2}

(n - 2)^{3} = 512

32 t - 2 t^{3} = 0

What are the solutions to the equation x+x^2-x^4+x^8=125 ?
The solutions to the equation x+x^2-x^4+x^8=125 are x\approx 1.83973…,x\approx-1.84635…
Find the zeros of x+x^2-x^4+x^8=125
The zeros of x+x^2-x^4+x^8=125 are x\approx 1.83973…,x\approx-1.84635…