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Let $\alpha,\beta$ be real roots of the quadratic equation $x^2-kx+k^2+k-5=0$.If $m,M$ are the minimum and maximum value of $\alpha^2+\beta^2$,then find $m+M.$


I calculated $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=k^2-2(k^2+k-5)=-k^2-2k+10$

I need to find the maximum and minimum value of this quadratic expression $-k^2-2k+10,$ which is a downward parabola.But as i have not been given the range of values of $k$,how should i find the maximum and minimum value of this quadratic expression $-k^2-2k+10?$

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  • $\begingroup$ Differentiatin wrt k $\endgroup$ – Archis Welankar Dec 31 '15 at 13:38
  • $\begingroup$ If the range of values of $k$ wasn't mentioned, it's probably safe to assume that $k$ can be any number in $\Bbb R$ $\endgroup$ – Kitegi Dec 31 '15 at 13:39
  • $\begingroup$ @Farnight then the question doesn't make any sense as the polynomial doesn't have a maximum $\endgroup$ – Kamil Jarosz Dec 31 '15 at 13:54
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Your delta must be greater than (or equal to) $0$:

$$\Delta\geqslant 0$$ $$k^2-4(k^2+k-5)\geqslant 0$$

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Hint: We don't always have real roots $\alpha$ and $\beta$. Use the discriminant to determine which values of $k$ we should consider.

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Hint the max value of real roots is when discriminant is $0$ and minimum when greater than $0$ so you get range fir k and get max and minimum values also we can differentiate if $f'(x)<0$ then maxima if $f'(x)>0$ then mibima

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