Question

4[x raised to power 2 +1/xraised to power 2]-[x+1/x]=60
Solve x

Answer 1

Final Answer : x = 4 or 1/4 or (√3 - 2 )or( -2-√3)

Q :
$4( {x}^{2} + \frac{1}{ {x}^{2} } ) - (x + \frac{1}{x} ) = 60$


Steps :
1) Let
$x + \frac{1}{x} = t \: \\ then \: \: {x}^{2} + \frac{1}{ {x}^{2} } = {t}^{2} - 2$

2) Since, t is
$4( {t}^{2} - 2) - t = 60 \\ = > 4 {t}^{2} - t - 68 = 0 \\ = > 4 {t}^{2} + 16t - 17t - 68 = 0 \\ = > 4t(t + 4) - 17(t + 4) = 0 \\ = (t + 4)(4t - 17) = 0 \\ = > t = - 4\: \: or \\ \: t \: = \frac{17}{4}$
3) When t = -4
$x + \frac{1}{x} = - 4 \\ = > {x}^{2} + 4x + 1 = 0 \\ = > x = \frac{ - 4 + \sqrt{12} }{2} or \: \frac{ - 4 - \sqrt{12} }{2} \\ = > x = -2 + \sqrt{3} or \: - 2 - \sqrt{3}$


4) When t = 17/4
$x + \frac{1}{x} = \frac{17}{4} \\ = > 4 {x}^{2} - 17x + 4 = 0 \\ = > 4 {x}^{2} - 16x - x + 4 = 0 \\ = > 4x(x - 4) - 1(x - 4) = 0 \\ = > (x - 4)(4x - 1) = 0 \\ = > x = 4 \: \: \: or \: x \: = \frac{1}{4}$



Therefore,
$x = 4 \: or \: \frac{1}{4} \: or \: - 2 - \sqrt{3} or \: \sqrt{3} - 2$