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  4. For all ' x' , x2 + 2ax + (10 - 3a) > 0, then the interval in which 'a' lies is

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Q. For all '$ x$' , $x^2 + 2ax + (10 - 3a) > 0,$ then the interval in which '$a$' lies is

IIT JEEIIT JEE 2004Complex Numbers and Quadratic Equations

Solution:

As we know, $ ax^2 + bx + c > 0 $ for all $ x \, \in \, R, $ iff $a > 0$ and $D < 0 $
Given equation is $ x^2 + 2ax + (10 - 3a) > 0, \, \forall \, x \, \in R $
Now, $ D < 0$
$\Rightarrow 4a^2 - 4 \, (10 - 3a ) < 0 $
$\Rightarrow 4 \, (a^2 + 3a - 10 ) < 0 $
$\Rightarrow \, (a + 5) \, (a + 2) < 0 $
$\Rightarrow a \, \in ( - 5, 2 )$