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  4. If x is rational and 4(x2 + (1/x2))+16(x + (1/x))-57=0, then the product of all possible values of x is

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Q. If $x$ is rational and $4\left(x^{2} + \frac{1}{x^{2}}\right)+16\left(x + \frac{1}{x}\right)-57=0,$ then the product of all possible values of $x$ is

NTA AbhyasNTA Abhyas 2022

Solution:

Given equation,
4 $\left(x^{2} + \frac{1}{x^{2}}\right)+16\left(x + \frac{1}{x}\right)-57=0$
Let, $x+\frac{1}{x}=y;$ $x^{2}+\frac{1}{x^{2}}=y^{2}-2$
$\Rightarrow $ $4y^{2}+16y-65=0$
$\Rightarrow $ $y=-\frac{1 3}{2}or \frac{5}{2}$
When, $y = \frac{5}{2}$
$x+\frac{1}{x}=\frac{5}{2}\Rightarrow x=2or\frac{1}{2}$
When, $y=-\frac{13}{2}$
$\Rightarrow $ $x+\frac{1}{x}=-13/2$
$\Rightarrow $ $2 x^{2} + 1 3 x + 2 = 0$
$\Rightarrow $ $x = \frac{- 1 3 \pm \sqrt{1 5 3}}{4}$
Since $x$ is rational, $x=2$ or $\frac{1}{2}$
Hence, their product is $1$ .