First slide

Theory of equations

Question

If $a, b, c \in R$ and $a + b + c = 0$ , the the quadratic equation $4 a x^{2} + 3 b x + 2 c = 0$ has

Moderate

A

one positive and one negative root
B

imaginary roots
C

real roots
D

none of these

Solution

Let $D$ be the discriminant of the given quadratic.
Then
$\begin{array}{l} D = 9 b^{2} - 32 a c \\ \Rightarrow D = 9 (- a - c)^{2} - 32 a c [∵ a + b + c = 0] \\ \Rightarrow D = 9 a^{2} + 9 c^{2} - 14 a c \\ \Rightarrow D = c^{2} \{9 {(\frac{a}{c})}^{2} - 14 (\frac{a}{c}) + 9\} = c^{2} \{{(\frac{3 a}{c} - \frac{7}{3})}^{2} + \frac{32}{9}\} > 0 \end{array}$
Hence, the roots are real.

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