What are the solutions to the equation x+x^2+x^4=2000 ?

The solutions to the equation x+x^2+x^4=2000 are x\approx 6.64453…,x\approx-6.65571…

Find the zeros of x+x^2+x^4=2000

The zeros of x+x^2+x^4=2000 are x\approx 6.64453…,x\approx-6.65571…

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x + x^{2} + x^{4} = 2000

x \approx 6.64453 \dots, x \approx - 6.65571 \dots

Solution steps

x + x^{2} + x^{4} = 2000

Move

2000

to the left side

x + x^{2} + x^{4} - 2000 = 0

Write in the standard form

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

x^{4} + x^{2} + x - 2000 = 0

Find one solution for

x^{4} + x^{2} + x - 2000 = 0

using Newton-Raphson:

x \approx 6.64453 \dots

Apply long division:

\frac{x ^{4} + x ^{2} + x - 2000}{x - 6.64453 \dots} = x^{3} + 6.64453 \dots x^{2} + 45.14980 \dots x + 300.99936 \dots

x^{3} + 6.64453 \dots x^{2} + 45.14980 \dots x + 300.99936 \dots \approx 0

Find one solution for

x^{3} + 6.64453 \dots x^{2} + 45.14980 \dots x + 300.99936 \dots = 0

using Newton-Raphson:

x \approx - 6.65571 \dots

Apply long division:

\frac{x ^{3} + 6.64453 \dots x ^{2} + 45.14980 \dots x + 300.99936 \dots}{x + 6.65571 \dots} = x^{2} - 0.01117 \dots x + 45.22421 \dots

x^{2} - 0.01117 \dots x + 45.22421 \dots \approx 0

Find one solution for

x^{2} - 0.01117 \dots x + 45.22421 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 6.64453 \dots, x \approx - 6.65571 \dots

x^{3} + x + 3 = 0

3 x^{4} - 8 x^{2} - 4 = 0

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

x^{3} - 5 x^{2} + 2 x - 10 = 0

4 x + 6 x^{3} - 2 x^{2} + 7 = 0

What are the solutions to the equation x+x^2+x^4=2000 ?
The solutions to the equation x+x^2+x^4=2000 are x\approx 6.64453…,x\approx-6.65571…
Find the zeros of x+x^2+x^4=2000
The zeros of x+x^2+x^4=2000 are x\approx 6.64453…,x\approx-6.65571…