First slide

Theory of equations

Question

lf a + b + c=0, then the roots of the equation $4 {ax}^{2} + 3 bx + 2 c = 0$ are

Moderate

A

equal
B

imaginary
C

real
D

None of these

Solution

Given, a, + b + c=0 ………………..(i)
and $4 {ax}^{2} + 3 bx + 2 c = 0$
$\begin{aligned} ∵ D = b^{2} - 4 a c \\ ∴ D = (3 b)^{2} - 4 (4 a) (2 c) \\ = 9 b^{2} - 32 a c \end{aligned}$
$\begin{array}{r} = 9 (a + c)^{2} - 32 a c [∵ from Eq. (i)] \\ = 9 (a - c)^{2} + 4 a c > 0 \end{array}$
Hence, roots are real.

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