First slide

Theory of equations

Question

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

Moderate

A

Equal
B

Imaginary
C

Real
D

None of these

Solution

We have $4 {ax}^{2} + 3 bx + 2 x = 0$ Let roots are $α$ and $β$
Let $D = B^{2} - 4 AC = 9 b^{2} - 4 (4 a) (2 c) = 9 b^{2} - 32 ac$
Given that, $(a + b + c) = 0 \Rightarrow b = - (a + c)$
Putting this value, we get
$= 9 {(a + c)}^{2} - 32 ac = 9 {(a - c)}^{2} + 4 ac .$
Hence roots are real.

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