Question

find the value of (x³-1/x³) if x/2 = 1/2x +1​

Answer 1

$\begin{gathered} \frac{x}{2} = \frac{1}{2x} + 1 \\ = > \frac{x}{2} - \frac{1}{2x} = 1 \\ = > \frac{ {x}^{2} - 1}{2x} = 1 \\ = > \frac{ {x}^{2} - 1}{x} = 2 \\ = > x - \frac{1}{x} = 2\end{gathered}$

Now, cubing throughout, we get,

$\begin{gathered} {(x - \frac{1}{x} )}^{3} = 8 \\ = > \: {x}^{3} - \frac{1}{ {x}^{3} } - 3(x - \frac{1}{x} ) = 8 \\ = > {x}^{3} - \frac{1}{ {x}^{3} } - 3 \times 2 = 8 \\ = > {x}^{3} - \frac{1}{ {x}^{3} } = 8 + 6 = 14\end{gathered} </p><p>$

Answer 2

Step-by-step explanation:

If the given statement is this, x/2 = 1/(2x) + 1 then,

x/2 - 1/(2x) = 1

(2x² - 1)/2x = 1

2x² - 1 = 2x

2x² - 2x - 1 = 0

x = [2 ± √(4 + 8)] / 4 = (2 ± 2√3)/2 = 1 ± √3

x³ = 1 ± 3√3 ± 3√3(1±√3) = 1 ± 3√3 ± 3√3 + 9

= 10 ± 6√3

1/x³ = 1/(10±6√3)

Multiply and divide by (10∓6√3) to rationalize

using formula (a+b)(a-b) = a² - b²

1/x³ = (10∓6√3)/(100-108) = -(10∓6√3)/8 = (5±3√3)/4

x³ - 1/x³ = 10 ± 6√3 - {(5±3√3)/4}

= (40 ± 24√3 - 5 ∓ 3√3)/4

= (35 ± 21√3)/4

If the given statement is this, x/2 = 1/(2x + 1) then,

2x² + x = 2

2x² + x - 2 = 0

x = [-1 ± √(1+16)]/4 = [-1 ± √17]/4

and rest of the calculation will take place in a similar manner