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If x is real and the roots of the equati...

If `x` is real and the roots of the equation `a x^2+b x+c=0` are imaginary, then prove tat `a^2x^2+a b x+a c` is always positive.

Text Solution

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The correct Answer is:
` a in ( infty), -2)`
Given, roots of `ax^(2) + bx + c = 0` are imaginary . Hence,
` b^(2) - 4ac lt 0 ` (1)
Let us consider `f(x) = a^(2) x^(2) + abx + ac `. Her, coefficient of f(x)
is `a^(2)` which is + ve, which makes graph concave upward. Also,
` D = (ab)^(2) - 4a^(2) (ac) = a^(2) (b^(2) - 4ac) lt 0`
Hence, `f(x) gt 0 , AA x in ` R .
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