First slide

Theory of expressions

Question

If $a$ and $b$ are the non-zero distinct roots of $x^{2} + a x + b = 0,$ then the least value of $x^{2} + a x + b$ , is

Moderate

A

$\frac{2}{3}$
B

$\frac{9}{4}$
C

$- \frac{9}{4}$
D

1

Solution

Since $a, b$ are roots of $x^{2} + a x + b = 0$ . Therefore,
$a^{2} + a^{2} + b = 0$ and, $b^{2} + a b + b = 0$
$\begin{array}{l} \Rightarrow b = - 2 a^{2} and b + a + 1 = 0 \\ \Rightarrow - 2 a^{2} + a + 1 = 0 \\ \Rightarrow 2 a^{2} - a - 1 = 0 \Rightarrow a = 1 or, a = - 1 / 2 . \end{array}$
Now,
$a = 1, \Rightarrow b = - 2 [∵ b + a + 1 = 0]$
and, $a = - 1 / 2 \Rightarrow b - - 1 / 2$
But, $a \neq b .$ Therefore $, a = 1, b = - 2 .$
$∴$ Least value of $x^{2} + a x + b$ is $- \frac{D}{4}$
i.e. $- (\frac{a^{2} - 4 b}{4}) = - (\frac{1 + 8}{4}) = - \frac{9}{4}$

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