The equation \( \Large x \left(\frac{3}{4}log_{2}x\right)^{2}+ \left(log_{2}x\right)
-\frac{5}{4}=\sqrt{2} \) has
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A) at least one real solution |
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B) exactly three real solution |
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C) exactly one irrational solution. |
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D) all of the above |
Correct Answer:
| C) exactly one irrational solution. |
For given equation to be meaningful we must have x > 0 for x > 0, the given equation can be written as \( \Large \frac{3}{4} \left(log_{2}x\right)^{2}+log_{2}x-\frac{5}{4} \)
= \( \Large logx\sqrt{2}=\frac{1}{2}logx^{2} \)
Putting \( \Large t=log_{2}x \) so that \( \Large logx^{2}=\frac{1}{t} \)
\( \Large \therefore \frac{3}{4}t^{2}+t-\frac{5}{4}=\frac{1}{2} \left(\frac{1}{t}\right) \)
=> \( \Large 3t^{3}+4t^{2}-5t-2=0 \)
=> \( \Large \left(t-1\right) \left(t+2\right) \left(3t+1\right)=0 \)
\( \Large log_{2}x=t=1, -2, -\frac{1}{3} \)
=> \( \Large x=2, 2^{-2}, 2^{\frac{1}{3}}\ or\ x=2, \frac{1}{4}, \frac{1}{2^{\frac{1}{3}}} \)
Thus, the given equation has exactly three real solutions out of which exactly one is irrational i.e. \( \Large \frac{1}{2^{\frac{1}{3}}} \)
Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations
