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If ax^(2) + bx + c = 0 , a , b, c in ...

If `ax^(2) + bx + c = 0 , a , b, c in` R has no real roots, and if ` c lt ` 0, the which of the following is ture ? (a) a` lt` 0 (b) a + b + c `gt` 0 (c) ` a +b +c lt ` 0

Text Solution

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Let `f(x) = ax^(2) + bx + c ` . Since f(x) has no real roots either `f(x) gt 0`
or `f(x) lt 0 ` for all x `in` R . Since `f(0) = c lt 0 `, we get ` f(x) lt 0 0, AA x in R` .
Therefore, a`lt 0 ` as the parabola y = f(x) opens downwards . Also
`f(1) = a + b + c lt 0 ` .
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