First slide

Theory of equations

Question

For the quadratic equation $x^{2} + 2 (a + 1) x + 9 a - 5 = 0$ , which of the following is/are true?

Moderate

A

If 2<a<5, then roots are of opposite sign.
B

If a < 0, then roots are of opposite sign.
C

lf a > 7, then both roots are negative.
D

If $2 \leq a \leq 5$ , then roots are unreal.

Solution

Given equation is
$\begin{array}{ll} x^{2} + 2 (a + 1) x + 9 a - 5 = 0 \\ D = 4 (a + 1)^{2} - 4 (9 a - 5) = 4 (a - 1) (a - 6) \\ ∴ D \geq 0 \Rightarrow a \leq 1 or a \geq 6 \Rightarrow roots are real \end{array}$
lf a < 0, then 9a - 5 < 0. Hence, the products of roots is less than 0. So, the roots are of opposite sign .lf a > 7 , then sum of roots is -2(a + 1) < 0. Product of roots is greater than 0.

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