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  4. If the quadratic polynomial f(x)=(a-3) x2-2 a x+3 a-7 ranges from [-1, ∞) for every x ∈ R, then the value of a lies in

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Q. If the quadratic polynomial $f(x)=(a-3) x^2-2 a x+3 a-7$ ranges from $[-1, \infty)$ for every $x \in R$, then the value of a lies in

Complex Numbers and Quadratic Equations

Solution:

$ f(x)=(9-3) x^2-2 a x+3 a-7$
Since range of $f(x) \in[-1, \infty) \forall x \in R$
Hence range of $(a-3) x^2-2 a x+3 a-6 \in[0, \infty)$
$\therefore a-3>0$ and $D=0$
on solving we get $a =6$ only