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  4. If the roots of the equation x2-8x+a2-60=0 are real, then the value of a is

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Q. If the roots of the equation $ {{x}^{2}}-8x+{{a}^{2}}-60=0 $ are real, then the value of $ a $ is

Rajasthan PETRajasthan PET 2003

Solution:

$ \because $ Roots of the equation
$ {{x}^{2}}-8x+{{a}^{2}}-6a=0 $ are real.
$ \therefore $ $ {{B}^{2}}-4AC\ge 0 $
$ \Rightarrow $ $ {{(8)}^{2}}-4(1)({{a}^{2}}-6a)\ge 0 $
$ \Rightarrow $ $ 64-4{{a}^{2}}+24a\ge 0 $
$ \Rightarrow $ $ 16-{{a}^{2}}+6a\ge 0 $
$ \Rightarrow $ $ {{a}^{2}}-6a-16\le 0 $
$ \Rightarrow $ $ (a+2)(a-8)\le 0 $
$ \Rightarrow $ $ -2\le a\le 8 $