What are the solutions to the equation (x-2)^5=8(x-2)^3 ?

The solutions to the equation (x-2)^5=8(x-2)^3 are x=2,x=2(1+sqrt(2)),x=2(1-sqrt(2))

Find the zeros of (x-2)^5=8(x-2)^3

The zeros of (x-2)^5=8(x-2)^3 are x=2,x=2(1+sqrt(2)),x=2(1-sqrt(2))

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

x = 2, x = 2 (1 + 2), x = 2 (1 - 2)

Decimal Notation

x = 2, x = 4.82842 \dots, x = - 0.82842 \dots

Solution steps

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

Expand

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

Expand

8 (x - 2)^{3} : 8 x^{3} - 48 x^{2} + 96 x - 64

x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32 = 8 x^{3} - 48 x^{2} + 96 x - 64

Subtract

8 x^{3} - 48 x^{2} + 96 x - 64

from both sides

x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32 - (8 x^{3} - 48 x^{2} + 96 x - 64) = 8 x^{3} - 48 x^{2} + 96 x - 64 - (8 x^{3} - 48 x^{2} + 96 x - 64)

Simplify

x^{5} - 10 x^{4} + 32 x^{3} - 32 x^{2} - 16 x + 32 = 0

Factor

x^{5} - 10 x^{4} + 32 x^{3} - 32 x^{2} - 16 x + 32 : (x - 2)^{3} (x^{2} - 4 x - 4)

(x - 2)^{3} (x^{2} - 4 x - 4) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

x - 2 = 0 : x = 2

Solve

x^{2} - 4 x - 4 = 0 : x = 2 (1 + 2), x = 2 (1 - 2)

The solutions are

x = 2, x = 2 (1 + 2), x = 2 (1 - 2)

2 x^{3} + 3 x^{2} - 5 x + 74 = 3 x^{2} + 8 x - 6

8 t^{2} - 4 t^{3} = - 60 t

(x^{2} - 4) \cdot (x^{2} - 3 x + 1) = 0

x^{8} = \frac{1}{2}

x^{3} - 4 x^{2} - 6 x + 9 = 0

What are the solutions to the equation (x-2)^5=8(x-2)^3 ?
The solutions to the equation (x-2)^5=8(x-2)^3 are x=2,x=2(1+sqrt(2)),x=2(1-sqrt(2))
Find the zeros of (x-2)^5=8(x-2)^3
The zeros of (x-2)^5=8(x-2)^3 are x=2,x=2(1+sqrt(2)),x=2(1-sqrt(2))