First slide

Theory of equations

Question

$If a + b + c = 0, then the roots of the equation (c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + (b^{2} - a c) = 0 are$

Moderate

A

real and equal
B

imaginary
C

real and unequal
D

None of these

Solution

$\begin{array}{l} D = 4 [{(a^{2} - b c)}^{2} - (b^{2} - a c) (c^{2} - a b)] \\ = 4 [a^{4} + b^{2} c^{2} - 2 a^{2} b c - \{b^{2} c^{2} - a c^{3} - a b^{3} + a^{2} b c\}] \\ = 4 a (a^{3} + b^{3} + c^{3} - 3 a b c) \\ D = 4 a (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - a c) = 0 \\ [as a + b + c = 0] \end{array}$
Hence the roots are equal

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