What are the solutions to the equation x^5+(2-x)^5=82 ?

The solutions to the equation x^5+(2-x)^5=82 are x\approx-0.41421…,x\approx 2.41421…

Find the zeros of x^5+(2-x)^5=82

The zeros of x^5+(2-x)^5=82 are x\approx-0.41421…,x\approx 2.41421…

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x^{5} + (2 - x)^{5} = 82

x \approx - 0.41421 \dots, x \approx 2.41421 \dots

Solution steps

x^{5} + (2 - x)^{5} = 82

Expand

x^{5} + (2 - x)^{5} : 10 x^{4} - 40 x^{3} + 80 x^{2} - 80 x + 32

10 x^{4} - 40 x^{3} + 80 x^{2} - 80 x + 32 = 82

Move

82

to the left side

10 x^{4} - 40 x^{3} + 80 x^{2} - 80 x - 50 = 0

Find one solution for

10 x^{4} - 40 x^{3} + 80 x^{2} - 80 x - 50 = 0

using Newton-Raphson:

x \approx - 0.41421 \dots

Apply long division:

\frac{10 x ^{4} - 40 x ^{3} + 80 x ^{2} - 80 x - 50}{x + 0.41421 \dots} = 10 x^{3} - 44.14213 \dots x^{2} + 98.28427 \dots x - 120.71067 \dots

10 x^{3} - 44.14213 \dots x^{2} + 98.28427 \dots x - 120.71067 \dots \approx 0

Find one solution for

10 x^{3} - 44.14213 \dots x^{2} + 98.28427 \dots x - 120.71067 \dots = 0

using Newton-Raphson:

x \approx 2.41421 \dots

Apply long division:

\frac{10 x ^{3} - 44.14213 \dots x ^{2} + 98.28427 \dots x - 120.71067 \dots}{x - 2.41421 \dots} = 10 x^{2} - 20 x + 50.00000 \dots

10 x^{2} - 20 x + 50.00000 \dots \approx 0

Find one solution for

10 x^{2} - 20 x + 50.00000 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx - 0.41421 \dots, x \approx 2.41421 \dots

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

(x^{2} + 1)^{3} = 0

2 x^{3} - 5 x^{2} + 1 = 0

(x + 2)^{4} + (x + 4)^{4} = 2

x^{5} + 5 x^{2} + 3 x - 9 = 0

What are the solutions to the equation x^5+(2-x)^5=82 ?
The solutions to the equation x^5+(2-x)^5=82 are x\approx-0.41421…,x\approx 2.41421…
Find the zeros of x^5+(2-x)^5=82
The zeros of x^5+(2-x)^5=82 are x\approx-0.41421…,x\approx 2.41421…