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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Solution of quadratic equations and Nature of roots

  • question_answer
    A real root of the equation \[{{\log }_{4}}\{{{\log }_{2}}(\sqrt{x+8}-\sqrt{x})\}=0\] is [AMU 1999]

    A) 1

    B) 2

    C) 3

    D) 4

    Correct Answer: A

    Solution :

    \[{{\log }_{4}}\left\{ \,{{\log }_{2}}(\sqrt{x+8}-\sqrt{x}) \right\}=0\] \[\Rightarrow {{4}^{0}}={{\log }_{2}}\left( \sqrt{x+8}-\sqrt{x} \right)\] \[\Rightarrow {{2}^{1}}=\sqrt{x+8}-\sqrt{x}\] \[\Rightarrow 4=x+8+x-2\sqrt{{{x}^{2}}+8x}\] \[\Rightarrow 2\sqrt{{{x}^{2}}+8x}=2x+4\] \[\Rightarrow {{x}^{2}}+8x={{x}^{2}}+4+4x\] \[\Rightarrow 4x=4\] \[\Rightarrow x=1\].


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