If the equation a x 2 + 2 b x - 3 c = 0 has no real roots and 3 c 4 a + b , then

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Question: Answered & Verified by Expert

If the equation

a x^{2} + 2 b x - 3 c = 0

has no real roots and

\frac{3 c}{4} < a + b,

then

MathematicsQuadratic EquationJEE Main

Options:

A $c < 0$
B $c > 0$
C $c = 0$
D $a + 2 b - 3 c < 0$

Solution:

1183 Upvotes Verified Answer

The correct answer is:

c < 0

$a x^{2} + 2 b x - 3 c = 0$ has no real roots

$f (2) = 4 a + 4 b - 3 c$
$= 4 (a + b - \frac{3 c}{4}) > 0$ (given)
So, $f (x) > 0 \forall x$
$\Rightarrow f (0) > 0 \Rightarrow - 3 c > 0 \Rightarrow c < 0$

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