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10th Class Mathematics Quadratic Equations Question Bank Quadratic Equations

  • question_answer
    If the roots of the equation \[~{{x}^{2}}+2cx+ab=0\]are real and unequal, then the equation \[{{x}^{2}}-2(a+b)x+{{a}^{2}}+{{b}^{2}}+2{{c}^{2}}=0\] has _____ roots.                         

    A)  Real                           

    B)  Equal              

    C)              No real

    D)  Can't be determined

    Correct Answer: C

    Solution :

    The given equation \[{{x}^{2}}+2cx+ab=0\] has real and unequal roots.                            \[\Rightarrow \]            \[D={{(2c)}^{2}}-4ab>0\] \[\Rightarrow \]       \[4{{c}^{2}}-4ab>0\] \[\Rightarrow \] \[{{c}^{2}}-ab>0\]               Now, the equation                            \[{{x}^{2}}-2(a+b)x+{{a}^{2}}+{{b}^{2}}+2{{c}^{2}}=0\] \[\therefore \] \[D={{(-2(a+b))}^{2}}-4({{a}^{2}}+{{b}^{2}}+2{{c}^{2}})\] \[=4{{a}^{2}}+4{{b}^{2}}+8ab-4{{a}^{2}}-4{{b}^{2}}-8{{c}^{2}}\] \[=8ab-8{{c}^{2}}=8\,(ab-{{c}^{2}})<0\] \[(\therefore \,\,{{c}^{2}}-ab>0)\] Hence, the equation has no real roots.         


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