The value of a for which the sum of the squares of the roots of the equation \[{x^2} - \left( {a - 2} \right)x - a - 1 = 0\]
assume the least value is :
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Official solution
Given quadratic equation,
\[{x^2} - \left( {a - 2} \right)x - a - 1 = 0\]
Let \[\alpha \] and \[\beta \] are the roots of the equation.
\[ \therefore \] \[\alpha \] + \[\beta \] = a−2
and \[\alpha \] β=−a−1Nowα2+β2=(α+β)2−2αβ⇒{\alpha ^2} + {\beta ^2} = {\left( {a - 2} \right)^2} + 2\left( {a + 1} \right) (
\] \Rightarrow α2+β2=a2−2a+6⇒α2+β2=(a−1)2+5⇒The value ofα2+β2will be minimum, whena−1= 0⇒a=1