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(k+1)^2-4(3k^2-2)(k^3)=0

Solution

(k + 1)^{2} - 4 (3 k^{2} - 2) (k^{3}) = 0

k \approx - 0.39166 \dots, k \approx 1.00000 \dots, k \approx - 0.81317 \dots

Solution steps

(k + 1)^{2} - 4 (3 k^{2} - 2) (k^{3}) = 0

Expand

(k + 1)^{2} - 4 (3 k^{2} - 2) (k^{3}) : k^{2} + 2 k + 1 - 12 k^{5} + 8 k^{3}

k^{2} + 2 k + 1 - 12 k^{5} + 8 k^{3} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 12 k^{5} + 8 k^{3} + k^{2} + 2 k + 1 = 0

Find one solution for

- 12 k^{5} + 8 k^{3} + k^{2} + 2 k + 1 = 0

using Newton-Raphson:

k \approx - 0.39166 \dots

Apply long division:

\frac{- 12 k ^{5} + 8 k ^{3} + k ^{2} + 2 k + 1}{k + 0.39166 \dots} = - 12 k^{4} + 4.70003 \dots k^{3} + 6.15914 \dots k^{2} - 1.41234 \dots k + 2.55317 \dots

- 12 k^{4} + 4.70003 \dots k^{3} + 6.15914 \dots k^{2} - 1.41234 \dots k + 2.55317 \dots \approx 0

Find one solution for

- 12 k^{4} + 4.70003 \dots k^{3} + 6.15914 \dots k^{2} - 1.41234 \dots k + 2.55317 \dots = 0

using Newton-Raphson:

k \approx 1.00000 \dots

Apply long division:

\frac{- 12 k ^{4} + 4.70003 \dots k ^{3} + 6.15914 \dots k ^{2} - 1.41234 \dots k + 2.55317 \dots}{k - 1} = - 12 k^{3} - 7.29996 \dots k^{2} - 1.14082 \dots k - 2.55317 \dots

- 12 k^{3} - 7.29996 \dots k^{2} - 1.14082 \dots k - 2.55317 \dots \approx 0

Find one solution for

- 12 k^{3} - 7.29996 \dots k^{2} - 1.14082 \dots k - 2.55317 \dots = 0

using Newton-Raphson:

k \approx - 0.81317 \dots

Apply long division:

\frac{- 12 k ^{3} - 7.29996 \dots k ^{2} - 1.14082 \dots k - 2.55317 \dots}{k + 0.81317 \dots} = - 12 k^{2} + 2.45816 \dots k - 3.13974 \dots

- 12 k^{2} + 2.45816 \dots k - 3.13974 \dots \approx 0

Find one solution for

- 12 k^{2} + 2.45816 \dots k - 3.13974 \dots = 0

using Newton-Raphson:

No Solution for

k \in R

The solutions are

k \approx - 0.39166 \dots, k \approx 1, k \approx - 0.81317 \dots

8 x = x^{4}

b^{3} = 343

s^{3} + 6 s^{2} + 12 s + 8 = 0

0 = m^{3} + 6 m^{2} + 9 m

27 y^{3} - 27 y^{2} + 9 y - 1 = 0