What are the solutions to the equation (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3 ?

The solutions to the equation (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3 are x\approx 0.84081…

Find the zeros of (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3

The zeros of (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3 are x\approx 0.84081…

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(x + 1)^{3} + (x + 2)^{3} + (x + 3)^{3} + (x + 4)^{3} = (x + 5)^{3}

x \approx 0.84081 \dots

Solution steps

(x + 1)^{3} + (x + 2)^{3} + (x + 3)^{3} + (x + 4)^{3} = (x + 5)^{3}

Expand

(x + 1)^{3} + (x + 2)^{3} + (x + 3)^{3} + (x + 4)^{3} : 4 x^{3} + 30 x^{2} + 90 x + 100

Expand

(x + 5)^{3} : x^{3} + 15 x^{2} + 75 x + 125

4 x^{3} + 30 x^{2} + 90 x + 100 = x^{3} + 15 x^{2} + 75 x + 125

Subtract

x^{3} + 15 x^{2} + 75 x + 125

from both sides

4 x^{3} + 30 x^{2} + 90 x + 100 - (x^{3} + 15 x^{2} + 75 x + 125) = x^{3} + 15 x^{2} + 75 x + 125 - (x^{3} + 15 x^{2} + 75 x + 125)

Simplify

3 x^{3} + 15 x^{2} + 15 x - 25 = 0

Find one solution for

3 x^{3} + 15 x^{2} + 15 x - 25 = 0

using Newton-Raphson:

x \approx 0.84081 \dots

Apply long division:

\frac{3 x ^{3} + 15 x ^{2} + 15 x - 25}{x - 0.84081 \dots} = 3 x^{2} + 17.52244 \dots x + 29.73310 \dots

3 x^{2} + 17.52244 \dots x + 29.73310 \dots \approx 0

Find one solution for

3 x^{2} + 17.52244 \dots x + 29.73310 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 0.84081 \dots

x^{3} - 3 x - 1 = 0

4 x^{4} - 36 x^{2} = 0

x^{3} - 2 x^{2} - 9 x + 18 = 0

x^{3} - 13 x + 12 = 0

2 x^{3} - 3 x^{2} = 9

What are the solutions to the equation (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3 ?
The solutions to the equation (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3 are x\approx 0.84081…
Find the zeros of (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3
The zeros of (x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3=(x+5)^3 are x\approx 0.84081…