If-x-x-1-3-4-x-1-3-10-Find-x-1-3-1-x-1-3-3-

Question Number 213791 by hardmath last updated on 16/Nov/24

If x - \sqrt[3]{x} - \frac{4}{\sqrt[3]{x}} = 10

Find \sqrt[3]{x} - \frac{1}{\sqrt[3]{x}} + 3 = ?

Commented by muallimRiyoziyot last updated on 19/Nov/24

x - 8 = \sqrt[3]{x} + 2 + \frac{4}{\sqrt[3]{x}}

(\sqrt[3]{x} - 2) (\sqrt[3]{x^{2}} + 2 \sqrt[3]{x} + 4) = \frac{\sqrt[3]{x^{2}} + 2 \sqrt[3]{x} + 4}{\sqrt[3]{x}}

\sqrt[3]{x} - 2 = \frac{1}{\sqrt[3]{x}} \to \sqrt[3]{x} - \frac{1}{\sqrt[3]{x}} = 2

Commented by muallimRiyoziyot last updated on 19/Nov/24

2 + 3 = 5

Answered by Ghisom last updated on 17/Nov/24

x^{4 / 3} - x^{2 / 3} - 10 x^{1 / 3} - 4 = 0

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

x^{1 / 3} - \frac{1}{x^{1 / 3}} + 3 = y

x^{2 / 3} - (y - 3) x^{1 / 3} - 1 = 0

y - 3 = 2 \Rightarrow y = 5

Answered by a.lgnaoui last updated on 17/Nov/24

posons z =^{3} \sqrt{x}

z^{3} - z - \frac{4}{z} = 10

z^{4} - z^{2} - 10 z - 4 = 0 (1)

z^{2} + 3 z - 1 = ?

(1) {[{(z^{2} - \frac{1}{2})}^{2} - \frac{17}{4}]}^{2} = 100 (z^{2} - \frac{1}{2}) + 50

{(k^{2} - \frac{17}{4})}^{2} = 100 k^{2} + 50 (X = z^{2} - \frac{1}{2})

X^{4} - \frac{17}{2} X^{2} - 100 X^{2} + \frac{17^{2}}{4^{2}} - 50 = 0

k^{4} - (\frac{217}{2}) k - \frac{511}{4^{2}} = 0

217^{2} + 511 / 64 = \frac{217^{2} \times 4 + 511}{64}

108, 647

X = \frac{217}{4} + \frac{108.647}{2} = \frac{217.294 + 217}{4}

= \frac{434.294}{4} = 108.573

z^{2} = 109.073 z = 10, 443

3 \sqrt{x} = 10, 443

^{3} \sqrt{x} - 1 /^{3} \sqrt{x} + 3 = 5, 19

Commented by Ghisom last updated on 17/Nov/24

this is wrong

Commented by a.lgnaoui last updated on 17/Nov/24

\sqrt{x} = \frac{10, 443}{3}

\sqrt{x} - \frac{1}{\sqrt{x}} + 3 = 5, 193 (there is no unique

solution)