What are the solutions to the equation (x-2)^4-(x-2)^2-6=0 ?

The solutions to the equation (x-2)^4-(x-2)^2-6=0 are x=-sqrt(3)+2,x=sqrt(3)+2

Find the zeros of (x-2)^4-(x-2)^2-6=0

The zeros of (x-2)^4-(x-2)^2-6=0 are x=-sqrt(3)+2,x=sqrt(3)+2

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(x - 2)^{4} - (x - 2)^{2} - 6 = 0

x = - 3 + 2, x = 3 + 2

Decimal Notation

x = 0.26794 \dots, x = 3.73205 \dots

Solution steps

(x - 2)^{4} - (x - 2)^{2} - 6 = 0

Expand

(x - 2)^{4} - (x - 2)^{2} - 6 : x^{4} - 8 x^{3} + 23 x^{2} - 28 x + 6

x^{4} - 8 x^{3} + 23 x^{2} - 28 x + 6 = 0

Find one solution for

x^{4} - 8 x^{3} + 23 x^{2} - 28 x + 6 = 0

using Newton-Raphson:

x \approx 0.26794 \dots

Apply long division:

\frac{x ^{4} - 8 x ^{3} + 23 x ^{2} - 28 x + 6}{x - 0.26794 \dots} = x^{3} - 7.73205 \dots x^{2} + 20.92820 \dots x - 22.39230 \dots

x^{3} - 7.73205 \dots x^{2} + 20.92820 \dots x - 22.39230 \dots \approx 0

Find one solution for

x^{3} - 7.73205 \dots x^{2} + 20.92820 \dots x - 22.39230 \dots = 0

using Newton-Raphson:

x \approx 3.73205 \dots

Apply long division:

\frac{x ^{3} - 7.73205 \dots x ^{2} + 20.92820 \dots x - 22.39230 \dots}{x - 3.73205 \dots} = x^{2} - 4 x + 6

x^{2} - 4 x + 6 \approx 0

Find one solution for

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

using Newton-Raphson:

No Solution for

x \in R

Approximate to algebraic form

x \approx 0.26794 \dots : x = - 3 + 2

x \approx 3.73205 \dots : x = 3 + 2

x = - 3 + 2, x = 3 + 2

The solutions are

x = - 3 + 2, x = 3 + 2

5 x^{3} + 5 x^{2} - 4 x - 4

n^{3} - n = 3 n^{2} - 3

x^{5} + 4 + 4 i = 0

4 x^{4} + 15 x^{2} - 4 = 0

4 x^{3} - 20 x^{2} - 9 x + 45 = 0

What are the solutions to the equation (x-2)^4-(x-2)^2-6=0 ?
The solutions to the equation (x-2)^4-(x-2)^2-6=0 are x=-sqrt(3)+2,x=sqrt(3)+2
Find the zeros of (x-2)^4-(x-2)^2-6=0
The zeros of (x-2)^4-(x-2)^2-6=0 are x=-sqrt(3)+2,x=sqrt(3)+2