If $m,M$ are the minimum and maximum value of $\alpha^2+\beta^2$,then find $m+M.$

Question

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 $-k^2-2k+10?$

Answer 1

Your delta must be greater than (or equal to) $0$:

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

Answer 2

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

Answer 3

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