First slide

Theory of equations

Question

$If a + b + c = 0 then the roots of the equation 4 a x^{2} + 3 b x + 2 c = 0 where a, b, c \in R are$

Moderate

A

real and distinct
B

imaginary
C

real and equal
D

infinite

Solution

$For given equation 4 a x^{2} + 3 b x + 2 c = 0$
$we have D = (3 b)^{2} - 4 (4 a) (2 c)$
$\begin{array}{l} = 9 b^{2} - 32 a c \\ = 9 (- a - c)^{2} - 32 a c \\ = 9 a^{2} - 14 a c + 9 c^{2} \\ = 9 c^{2} ({(\frac{a}{c})}^{2} - \frac{14}{9} \frac{a}{c} + 1) \\ = 9 c^{2} ({(\frac{a}{c} - \frac{7}{9})}^{2} - \frac{49}{81} + 1) \end{array}$
which is always positive
Hence roots are real and distinct.

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