If and are the roots of the equation x 2 + b x + c = 0 , where c 0 b , then [2000]

Question

If $α$ and $β$ $(α < β)$ are the roots of the equation $x^{2} + b x + c = 0,$ where $c < 0 < b$ , then [2000]

Smart Achievers · Accepted Answer

(2)

$Given : c < 0 < b and α + β = - b \dots (i)$

$c < 0 < b$

$From (ii), c < 0 \Rightarrow α β < 0 \Rightarrow Either α is negative or β is negative and second quantity is positive.$

$From (i), b > 0 \Rightarrow - b < 0 \Rightarrow α + β < 0$

$\Rightarrow the sum is negative$

$\Rightarrow (Modulus of negative quantity) > (Modulus of positive quantity)$

$But given α < β . Therefore, it is clear that α is negative and β is positive$

$and modulus of α is greater than modulus of β$

$\Rightarrow α < 0 < β < | α |$

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