What are the solutions to the equation (z+2)^3=z^2+5z+2 ?

The solutions to the equation (z+2)^3=z^2+5z+2 are z\approx-3.48558…

Find the zeros of (z+2)^3=z^2+5z+2

The zeros of (z+2)^3=z^2+5z+2 are z\approx-3.48558…

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(z + 2)^{3} = z^{2} + 5 z + 2

z \approx - 3.48558 \dots

Solution steps

(z + 2)^{3} = z^{2} + 5 z + 2

Expand

(z + 2)^{3} : z^{3} + 6 z^{2} + 12 z + 8

z^{3} + 6 z^{2} + 12 z + 8 = z^{2} + 5 z + 2

Move

2

to the left side

z^{3} + 6 z^{2} + 12 z + 6 = z^{2} + 5 z

Move

5 z

to the left side

z^{3} + 6 z^{2} + 7 z + 6 = z^{2}

Move

z^{2}

to the left side

z^{3} + 5 z^{2} + 7 z + 6 = 0

Find one solution for

z^{3} + 5 z^{2} + 7 z + 6 = 0

using Newton-Raphson:

z \approx - 3.48558 \dots

Apply long division:

\frac{z ^{3} + 5 z ^{2} + 7 z + 6}{z + 3.48558 \dots} = z^{2} + 1.51441 \dots z + 1.72137 \dots

z^{2} + 1.51441 \dots z + 1.72137 \dots \approx 0

Find one solution for

z^{2} + 1.51441 \dots z + 1.72137 \dots = 0

using Newton-Raphson:

No Solution for

z \in R

The solution is

z \approx - 3.48558 \dots

m^{3} - 10 m + 1 = 0

r^{4} - 6 0^{3} - 20600 r^{2} + 2166000 r + 32490000 = 0

x^{3} - x = \frac{1}{4} x^{2} - \frac{1}{4}

x^{3} + 3 x^{2} + 3 x - 7 = 0

8 x^{1} + x^{2} + 2 x^{3} + x^{4} = 8

What are the solutions to the equation (z+2)^3=z^2+5z+2 ?
The solutions to the equation (z+2)^3=z^2+5z+2 are z\approx-3.48558…
Find the zeros of (z+2)^3=z^2+5z+2
The zeros of (z+2)^3=z^2+5z+2 are z\approx-3.48558…