What are the solutions to the equation 8(x-5)^7=(x-5)^{10} ?

The solutions to the equation 8(x-5)^7=(x-5)^{10} are x=5,x=7,x=4+sqrt(3)i,x=4-sqrt(3)i

Find the zeros of 8(x-5)^7=(x-5)^{10}

The zeros of 8(x-5)^7=(x-5)^{10} are x=5,x=7,x=4+sqrt(3)i,x=4-sqrt(3)i

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8 (x - 5)^{7} = (x - 5)^{10}

x = 5, x = 7, x = 4 + 3 i, x = 4 - 3 i

Solution steps

8 (x - 5)^{7} = (x - 5)^{10}

Expand

8 (x - 5)^{7} : 8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000

Expand

(x - 5)^{10} : x^{10} - 50 x^{9} + 1125 x^{8} - 15000 x^{7} + 131250 x^{6} - 787500 x^{5} + 3281250 x^{4} - 9375000 x^{3} + 17578125 x^{2} - 19531250 x + 9765625

8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000 = x^{10} - 50 x^{9} + 1125 x^{8} - 15000 x^{7} + 131250 x^{6} - 787500 x^{5} + 3281250 x^{4} - 9375000 x^{3} + 17578125 x^{2} - 19531250 x + 9765625

Switch sides

x^{10} - 50 x^{9} + 1125 x^{8} - 15000 x^{7} + 131250 x^{6} - 787500 x^{5} + 3281250 x^{4} - 9375000 x^{3} + 17578125 x^{2} - 19531250 x + 9765625 = 8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000

Subtract

8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000

from both sides

x^{10} - 50 x^{9} + 1125 x^{8} - 15000 x^{7} + 131250 x^{6} - 787500 x^{5} + 3281250 x^{4} - 9375000 x^{3} + 17578125 x^{2} - 19531250 x + 9765625 - (8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000) = 8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000 - (8 x^{7} - 280 x^{6} + 4200 x^{5} - 35000 x^{4} + 175000 x^{3} - 525000 x^{2} + 875000 x - 625000)

Simplify

x^{10} - 50 x^{9} + 1125 x^{8} - 15008 x^{7} + 131530 x^{6} - 791700 x^{5} + 3316250 x^{4} - 9550000 x^{3} + 18103125 x^{2} - 20406250 x + 10390625 = 0

Factor

x^{10} - 50 x^{9} + 1125 x^{8} - 15008 x^{7} + 131530 x^{6} - 791700 x^{5} + 3316250 x^{4} - 9550000 x^{3} + 18103125 x^{2} - 20406250 x + 10390625 : (x - 5)^{7} (x - 7) (x^{2} - 8 x + 19)

(x - 5)^{7} (x - 7) (x^{2} - 8 x + 19) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x - 5 = 0 or x - 7 = 0 or x^{2} - 8 x + 19 = 0

Solve

x - 5 = 0 : x = 5

Solve

x - 7 = 0 : x = 7

Solve

x^{2} - 8 x + 19 = 0 : x = 4 + 3 i, x = 4 - 3 i

The solutions are

x = 5, x = 7, x = 4 + 3 i, x = 4 - 3 i

- 2 x^{5} - x^{4} - 2 x^{3} - x^{2} = 0

2 x^{3} - 12 x^{2} - 8 x + 48 = 0

x^{3} + 2 x^{2} = 16 x + 32

n^{3} = 256

x^{5} + 32 i = 0

What are the solutions to the equation 8(x-5)^7=(x-5)^{10} ?
The solutions to the equation 8(x-5)^7=(x-5)^{10} are x=5,x=7,x=4+sqrt(3)i,x=4-sqrt(3)i
Find the zeros of 8(x-5)^7=(x-5)^{10}
The zeros of 8(x-5)^7=(x-5)^{10} are x=5,x=7,x=4+sqrt(3)i,x=4-sqrt(3)i