Question

find the sum of all the rational roots of the equation (above in the picture)​

Answer 1

Step-by-step explanation:

${\LARGE{\mathbf{\underline{\red{Solution :→}}}}}$

Given :-

${\bold{\sf{: \: ⟹ \: \: \: \: \: _X \: \frac{3}{4} \: ( log_{2} \: x)² \: + \: log_{2} \: x - \: \frac{5}{4} \: = \: √2}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: \frac{3}{4}( log_{2} \: x)² \: + \: log_{2} \: x - \: \frac{5}{4} \: = \: log_X \: √2}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: \frac{3}{4}( log_{2} \: x)² \: + \: log_{2} \: x - \: \frac{5}{4} \: = \: \frac{1}{2log_2 \: x}}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: 3(log_2 \: x)³ \: + \: 4(log2 \: x)² \: - \: 5(log_2 \: x) \: - \: 2 \: = \: 0}}}$

${\bold{\sf{\pink{Put, \: log_2 \: x \: = \: y}}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: 3y³ + \: 4y² \: - \: 5y \: - \: 2 \: = \: 0}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: (y \: - \: 1) (y \: + \: 2) (3y \: + \: 1) \: = \: 0}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: y \: = 1, \: -2, \: \frac{-1}{3}}}} \\ \\ </u></p><p><u>[tex]{\bold{\sf{: \: ⟹ \: \: \: \: \: log_2 \: x = 1, \: -2, \: \frac{-1}{3}}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: x \: = \: 2, \: \frac{1}{2⅓}, \: \frac{-1}{4}}}}$

Hence, equation has exactly 3 solutions