First slide

Theory of equations

Question

$If a and b (\neq 0) are roots of the equation x^{2} + a x + b = 0, then the least value of x^{2} + a x + b (x \in R) is$

Moderate

A

$\frac{9}{4}$
B

$- \frac{1}{4}$
C

$\frac{1}{4}$
D

$- \frac{9}{4}$

Solution

$Since a and b are the roots of the equation x^{2} + a x + b = 0$
$Therefore, a + b = - a and a b = b$
$Now, a b = b \Rightarrow (a - 1) b = 0 \Rightarrow a = 1 (∵ b \neq 0)$
$Putting a = 1 in a + b = - a, we get b = - 2,$
$\Rightarrow x^{2} + a x + b = x^{2} + x - 2 = (x + 1 / 2)^{2} - 1 / 4 - 2 = (x + 1 / 2)^{2} - 9 / 4$
$which has minimum value - 9 / 4 .$

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