Hello-Solve-x-2-1-3-3-x-x-1-1-3-2-x-1-1-3-0-

Question Number 71406 by mind is power last updated on 15/Oct/19

Hello

Solve \sqrt[3]{x^{2}} - 3 \sqrt[3]{x (x - 1)} + 2 \sqrt[3]{x - 1} = 0

Commented by Prithwish sen last updated on 15/Oct/19

x^{2} + (x - 1) - 27 x (x - 1) + 18 x {(x - 1)}^{\frac{2}{3}} = 0

putting x = a^{3} and x - 1 = b^{3} \Rightarrow a^{3} = 1 + b^{3}

∴ a^{6} + {8 b}^{3} - {27 a}^{3} b^{3} + {18 a}^{3} b^{2} = 0

\Rightarrow 26 b^{6} - 18 b^{5} + 17 b^{3} - 18 b^{2} - 1 = 0

and I got a equation of 6 degrees .

Sir please comment .

Commented by MJS last updated on 15/Oct/19

a^{1 / 3} - b^{1 / 3} + c^{1 / 3} = 0

a^{1 / 3} + c^{1 / 3} = b^{1 / 3}

a + c + 3 a^{1 / 3} c^{1 / 3} (a^{1 / 3} + c^{1 / 3}) = b

3 a^{1 / 3} b^{1 / 3} c^{1 / 3} = b - a - c

27 a b c = {(b - a - c)}^{3}

a^{3} - b^{3} + c^{3} + 21 a b c - 3 (a^{2} b + a^{2} c + {a b}^{2} + {a c}^{2} + b^{2} c - {b c}^{2}) = 0

a = x^{2}

b = 27 x (x - 1)

c = 8 (x - 1)

x^{6} - \frac{19203}{4394} x^{5} + \frac{61719}{8788} x^{4} - \frac{92387}{17576} x^{3} + \frac{4299}{2197} x^{2} - \frac{840}{2197} x + \frac{64}{2197} = 0

I found no exact solution

x_{1} \approx . 172484

x_{2} \approx 1.72890

x_{3, 4} \approx . 230494 \pm . 209231 i

x_{5, 6} \approx 1.00395 \pm . 0115209 i

if we generally use \sqrt[3]{z} = abs {(z)}^{1 / 3} \times e^{i \frac{\arg (z)}{3}}

only x_{2} is a solution

if we use the common exception \sqrt[3]{- r} = - \sqrt[3]{r}

also x_{1} is a solution

the complex solutions are false

Commented by Prithwish sen last updated on 15/Oct/19

you just turn the equation in cubic form .

Great sir .

Commented by mind is power last updated on 15/Oct/19

thank you!, i have no better solution my bee we {can}^{'} t find exacte root^{'}

or its hard we need find function g (t) such that if we put x = g (t)

equation withe t is easier i try but no result

Commented by MJS last updated on 15/Oct/19

your equation is equivalent to mine

your equation is equivalent to mine

Commented by mind is power last updated on 15/Oct/19

i have did like you sir!

i have did like you sir!