What are the solutions to the equation (x-1)^6+(x-2)^6=1 ?

The solutions to the equation (x-1)^6+(x-2)^6=1 are x=1,x=2

Find the zeros of (x-1)^6+(x-2)^6=1

The zeros of (x-1)^6+(x-2)^6=1 are x=1,x=2

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(x - 1)^{6} + (x - 2)^{6} = 1

x = 1, x = 2

Solution steps

(x - 1)^{6} + (x - 2)^{6} = 1

Expand

(x - 1)^{6} + (x - 2)^{6} : 2 x^{6} - 18 x^{5} + 75 x^{4} - 180 x^{3} + 255 x^{2} - 198 x + 65

2 x^{6} - 18 x^{5} + 75 x^{4} - 180 x^{3} + 255 x^{2} - 198 x + 65 = 1

Move

1

to the left side

2 x^{6} - 18 x^{5} + 75 x^{4} - 180 x^{3} + 255 x^{2} - 198 x + 64 = 0

Factor

2 x^{6} - 18 x^{5} + 75 x^{4} - 180 x^{3} + 255 x^{2} - 198 x + 64 : (x - 1) (x - 2) (2 x^{4} - 12 x^{3} + 35 x^{2} - 51 x + 32)

(x - 1) (x - 2) (2 x^{4} - 12 x^{3} + 35 x^{2} - 51 x + 32) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x - 1 = 0 or x - 2 = 0 or 2 x^{4} - 12 x^{3} + 35 x^{2} - 51 x + 32 = 0

Solve

x - 1 = 0 : x = 1

Solve

x - 2 = 0 : x = 2

Solve

2 x^{4} - 12 x^{3} + 35 x^{2} - 51 x + 32 = 0 :

No Solution for

x \in R

The solutions are

x = 1, x = 2

- 6 x^{4} + 5 x^{3} + 6 x^{2} = 11 x + 5

(z^{2} + 7)^{2} + (z^{2} + 7) - 72 = 0

y, x = (y - 1)^{3}

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

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

What are the solutions to the equation (x-1)^6+(x-2)^6=1 ?
The solutions to the equation (x-1)^6+(x-2)^6=1 are x=1,x=2
Find the zeros of (x-1)^6+(x-2)^6=1
The zeros of (x-1)^6+(x-2)^6=1 are x=1,x=2