First slide

Theory of equations

Question

If the roots of $x^{2} + x + a = 0$ exceed a, then

Moderate

A

$2 < a < 3$
B

$a > 3$
C

$- 3 < a < 3$
D

$a < - 2$

Solution

If the roots of the quadratic equation ${ax}^{2} + bx + c = 0$ exceed a number k, then ${ak}^{2} + bk + c > 0$ if $a > 0, b^{2} - 4 ac \geq 0$ and sum of the roots $> 2 k$ therefore, if the roots of $x^{2} + x + a = 0$ exceed a number a, then $a^{2} + a + a > 0, 1 - 4 a \geq 0 and - 1 > 2 a$
$\Rightarrow a (a + 2) > 0, a \leq \frac{1}{4} and a < - \frac{1}{2}$
$\Rightarrow a > 0 or a < - 2, a < \frac{1}{4} and a < \frac{1}{2}$
Hence $a < - 2$

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