One or more options can be correct:

The condition for the roots of the quadratic equation $f (x) = a x^{2} + b x + c$ to be less than a real value $x_{0}$ is/are

$x_{0} > \frac{- b}{2 a}$

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$f (x_{0}) > 0$

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$f (x_{0}) < 0$

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$D \geq 0$

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Solution

The correct option is D
$D \geq 0$

Conditions for both roots to be less than a real value $x_{0}$ , can be visualized through graph

Here, $a = 1 (> 0), b = - 2 m, c = m^{2} - 1$

Observe that
$(i) f (x_{0}) > 0$

$(i i) x_{0} > \frac{- b}{2 a}$

Also for real roots to exist
$(i i i) b^{2} \geq 4 a c$

Hence, $A, C, D$ are right options.

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Q. If both the roots

α, β

of the quadratic equation

a x^{2} + b x + c = 0, a < 0

are less than a real number

k

such that

k > α > β

Considering

f (x) = a x^{2} + b x + c

, select the correct statement(s).

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One or more options can be correct: The condition for the roots of the quadratic equation f(x)=ax2+bx+c to be less than a real value x0 is/are

The correct option is D D≥0 Conditions for both roots to be less than a real value x0, can be visualized through graph Here, a=1(>0),b=−2m,c=m2−1 Observe that (i) f(x0)>0 (ii) x0>−b2a Also for real roots to exist (iii) b2≥4ac Hence, A,C,D are right options.

One or more options can be correct:

The condition for the roots of the quadratic equation $f (x) = a x^{2} + b x + c$ to be less than a real value $x_{0}$ is/are