For all ' x ' , x 2 + 2 a x + 10 - 3 a 0 , then the interval in which ' a ' lies is [2004]

Question

For all $' x', x^{2} + 2 a x + 10 - 3 a > 0,$ then the interval in which $' a'$ lies is [2004]

Smart Achievers · Accepted Answer

(2)

$f (x) = a x^{2} + b x + c has same sign as that of a if D < 0 .$

$Since x^{2} + 2 a x + 10 - 3 a > 0 \forall x$

$∴$ $D < 0 \Rightarrow 4 a^{2} - 4 (10 - 3 a) < 0$

$\Rightarrow a^{2} + 3 a - 10 < 0$

$\Rightarrow (a + 5) (a - 2) < 0 \Rightarrow a \in (- 5, 2)$

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