What are the solutions to the equation x^4+2x^3-x^2-6x-3=0 ?

The solutions to the equation x^4+2x^3-x^2-6x-3=0 are x=-(sqrt(5)-1)/2 ,x=(sqrt(5)+1)/2

Find the zeros of x^4+2x^3-x^2-6x-3=0

The zeros of x^4+2x^3-x^2-6x-3=0 are x=-(sqrt(5)-1)/2 ,x=(sqrt(5)+1)/2

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

x = - \frac{5 - 1}{2}, x = \frac{5 + 1}{2}

Decimal Notation

x = - 0.61803 \dots, x = 1.61803 \dots

Solution steps

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

Find one solution for

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

using Newton-Raphson:

x \approx - 0.61803 \dots

Apply long division:

\frac{x ^{4} + 2 x ^{3} - x ^{2} - 6 x - 3}{x + 0.61803 \dots} = x^{3} + 1.38196 \dots x^{2} - 1.85410 \dots x - 4.85410 \dots

x^{3} + 1.38196 \dots x^{2} - 1.85410 \dots x - 4.85410 \dots \approx 0

Find one solution for

x^{3} + 1.38196 \dots x^{2} - 1.85410 \dots x - 4.85410 \dots = 0

using Newton-Raphson:

x \approx 1.61803 \dots

Apply long division:

\frac{x ^{3} + 1.38196 \dots x ^{2} - 1.85410 \dots x - 4.85410 \dots}{x - 1.61803 \dots} = x^{2} + 3 x + 3

x^{2} + 3 x + 3 \approx 0

Find one solution for

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

using Newton-Raphson:

No Solution for

x \in R

Approximate to algebraic form

x \approx - 0.61803 \dots : x = - \frac{5 - 1}{2}

x \approx 1.61803 \dots : x = \frac{5 + 1}{2}

x = - \frac{5 - 1}{2}, x = \frac{5 + 1}{2}

The solutions are

x = - \frac{5 - 1}{2}, x = \frac{5 + 1}{2}

x^{4} - 17 x^{2} = - 16

x^{3} - 5 x - 1 = 0

4 x^{9} - 4 x = 0

4 (x - 6)^{3} (x^{2} + 7) = 0

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

What are the solutions to the equation x^4+2x^3-x^2-6x-3=0 ?
The solutions to the equation x^4+2x^3-x^2-6x-3=0 are x=-(sqrt(5)-1)/2 ,x=(sqrt(5)+1)/2
Find the zeros of x^4+2x^3-x^2-6x-3=0
The zeros of x^4+2x^3-x^2-6x-3=0 are x=-(sqrt(5)-1)/2 ,x=(sqrt(5)+1)/2