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Sign of Quadratic Expression

If a, b, c be...

Question

If a, b, c be distinct real and+ive numbers which are in H.P., then the roots of the equation $a x^{2} + 2 b x + c = 0$ are real and distinct. Is this statement true or not?

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Solution

Given $a, b, c \in H . P \Rightarrow b = \frac{2 a c}{a + c}$
Now discriminant of given quadratic, $Δ = 4 b^{2} - 4 a c = 4 [\frac{4 a^{2} c^{2}}{{(a + c)}^{2}} - a c]$ $= \frac{4 a c}{{(a + c)}^{2}} [4 a c - {(a + c)}^{2}] = - 4 a c \frac{{(a - c)}^{2}}{{(a + c)}^{2}}$
Since $a, b, c$ are distinct and +ve
, $Δ = - v e .$ Hence the roots are imaginary.

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