First slide

Theory of expressions

Question

The values of $' a'$ for which the roots of the equation $x^{2} + x + a = 0$ are real and exceed $‘ a ’$ are

Moderate

A

$0 < a < 1 / 4$
B

$a < 1 / 4$
C

$a < - 2$
D

$- 2 < a < 0$

Solution

Let $f (x) = x^{2} + x + a$ Both the roots of $f (x) = 0$ will exceed $a,$ if
(i) Discriminant > 0
(ii) a lies outside the roots i.e. $f (a) > 0$
(iii) $a < x$ coordinate or vertex
$∴ a < \frac{1}{4}, a^{2} + 2 a > 0$ and $a < - 1 / 2$
$\Rightarrow a < - 1 / 2$ and $a^{2} + 2 a > 0$
$\Rightarrow a < - 1 / 2$ and $a (a + 2) > 0$
$\Rightarrow a < - 1 / 2$ and $a + 2 < 0$ $[∵ a < 0]$
$\Rightarrow a < - 1 / 2$ and $a < - 2 \Rightarrow a < - 2$

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