What are the solutions to the equation r^5-5r=3 ?

The solutions to the equation r^5-5r=3 are r\approx-1.27568…,r\approx-0.61803…,r\approx 1.61803…

Find the zeros of r^5-5r=3

The zeros of r^5-5r=3 are r\approx-1.27568…,r\approx-0.61803…,r\approx 1.61803…

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r^{5} - 5 r = 3

r \approx - 1.27568 \dots, r \approx - 0.61803 \dots, r \approx 1.61803 \dots

Solution steps

r^{5} - 5 r = 3

Move

3

to the left side

r^{5} - 5 r - 3 = 0

Find one solution for

r^{5} - 5 r - 3 = 0

using Newton-Raphson:

r \approx - 1.27568 \dots

Apply long division:

\frac{r ^{5} - 5 r - 3}{r + 1.27568 \dots} = r^{4} - 1.27568 \dots r^{3} + 1.62736 \dots r^{2} - 2.07600 \dots r - 2.35168 \dots

r^{4} - 1.27568 \dots r^{3} + 1.62736 \dots r^{2} - 2.07600 \dots r - 2.35168 \dots \approx 0

Find one solution for

r^{4} - 1.27568 \dots r^{3} + 1.62736 \dots r^{2} - 2.07600 \dots r - 2.35168 \dots = 0

using Newton-Raphson:

r \approx - 0.61803 \dots

Apply long division:

\frac{r ^{4} - 1.27568 \dots r ^{3} + 1.62736 \dots r ^{2} - 2.07600 \dots r - 2.35168 \dots}{r + 0.61803 \dots} = r^{3} - 1.89371 \dots r^{2} + 2.79774 \dots r - 3.80510 \dots

r^{3} - 1.89371 \dots r^{2} + 2.79774 \dots r - 3.80510 \dots \approx 0

Find one solution for

r^{3} - 1.89371 \dots r^{2} + 2.79774 \dots r - 3.80510 \dots = 0

using Newton-Raphson:

r \approx 1.61803 \dots

Apply long division:

\frac{r ^{3} - 1.89371 \dots r ^{2} + 2.79774 \dots r - 3.80510 \dots}{r - 1.61803 \dots} = r^{2} - 0.27568 \dots r + 2.35168 \dots

r^{2} - 0.27568 \dots r + 2.35168 \dots \approx 0

Find one solution for

r^{2} - 0.27568 \dots r + 2.35168 \dots = 0

using Newton-Raphson:

No Solution for

r \in R

The solutions are

r \approx - 1.27568 \dots, r \approx - 0.61803 \dots, r \approx 1.61803 \dots

9 x^{4} = 25 x^{2} - 16

x, x^{3} + y^{3} - z^{3} = 4

x^{3} - 2 x - 5 = 0, (x = 2.0813)

(x - \frac{1}{3}) (x^{2} + x \frac{0.1}{3} + \frac{1 ^{2}}{3 ^{2}}) = 64

(x - 7)^{4} = (x - 7)^{5}

What are the solutions to the equation r^5-5r=3 ?
The solutions to the equation r^5-5r=3 are r\approx-1.27568…,r\approx-0.61803…,r\approx 1.61803…
Find the zeros of r^5-5r=3
The zeros of r^5-5r=3 are r\approx-1.27568…,r\approx-0.61803…,r\approx 1.61803…