Quadratic EquationHard
Question
The equation 
= √2 has
= √2 hasOptions
A.at least one real solution
B.exactly three real solution
C.exactly one irrational solution
D.complex roots
Solution
Given, = √2
⇒
(log2x)2 + log2 x = - 
= logx √2
⇒
(log2x)2 + log2 x = 
⇒ 3(log2x)3 + 4(log2x)2 - 5(log2x) - 2 = 0
Put, log2x = y
∴ 3y3 + 4y2 - 5y - 2 = 0
⇒ ( y - 1)( y - 2)(3y - 1) = 0
⇒ x = 1, - 2, - 1/3
⇒ log2x = 1, - 2, - 1/3
⇒ x = 2,
= √2 ⇒
(log2x)2 + log2 x = - = logx √2⇒
(log2x)2 + log2 x = ⇒ 3(log2x)3 + 4(log2x)2 - 5(log2x) - 2 = 0
Put, log2x = y
∴ 3y3 + 4y2 - 5y - 2 = 0
⇒ ( y - 1)( y - 2)(3y - 1) = 0
⇒ x = 1, - 2, - 1/3
⇒ log2x = 1, - 2, - 1/3
⇒ x = 2,
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