What are the solutions to the equation (x-1)^4+(x-7)^4=272 ?

The solutions to the equation (x-1)^4+(x-7)^4=272 are x=3,x=5,x=4+sqrt(55)i,x=4-sqrt(55)i

Find the zeros of (x-1)^4+(x-7)^4=272

The zeros of (x-1)^4+(x-7)^4=272 are x=3,x=5,x=4+sqrt(55)i,x=4-sqrt(55)i

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(x - 1)^{4} + (x - 7)^{4} = 272

x = 3, x = 5, x = 4 + 55 i, x = 4 - 55 i

Solution steps

(x - 1)^{4} + (x - 7)^{4} = 272

Expand

(x - 1)^{4} + (x - 7)^{4} : 2 x^{4} - 32 x^{3} + 300 x^{2} - 1376 x + 2402

2 x^{4} - 32 x^{3} + 300 x^{2} - 1376 x + 2402 = 272

Move

272

to the left side

2 x^{4} - 32 x^{3} + 300 x^{2} - 1376 x + 2130 = 0

Factor

2 x^{4} - 32 x^{3} + 300 x^{2} - 1376 x + 2130 : 2 (x - 3) (x - 5) (x^{2} - 8 x + 71)

2 (x - 3) (x - 5) (x^{2} - 8 x + 71) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x - 3 = 0 or x - 5 = 0 or x^{2} - 8 x + 71 = 0

Solve

x - 3 = 0 : x = 3

Solve

x - 5 = 0 : x = 5

Solve

x^{2} - 8 x + 71 = 0 : x = 4 + 55 i, x = 4 - 55 i

The solutions are

x = 3, x = 5, x = 4 + 55 i, x = 4 - 55 i

x^{4} + x^{2} - 1 = 0

x^{4} = - 1

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

z^{4} + 13 z^{2} + 36 = 0

x^{4} + x^{3} - 4 x^{2} + x + 1 = 0

What are the solutions to the equation (x-1)^4+(x-7)^4=272 ?
The solutions to the equation (x-1)^4+(x-7)^4=272 are x=3,x=5,x=4+sqrt(55)i,x=4-sqrt(55)i
Find the zeros of (x-1)^4+(x-7)^4=272
The zeros of (x-1)^4+(x-7)^4=272 are x=3,x=5,x=4+sqrt(55)i,x=4-sqrt(55)i