Question

1. Under which one of the following
conditions will the quadratic equation
x² + mx + 2 = 0 always have real roots ?

(a) 2√3 ≤m² <8
(b) √3 ≤m² <4
(©) m²≥8
(d) m²≤ 13​

Answer 1

$\textbf{Concept used:}$

$\text{The condition for which the quadratic equation ax^2+bx+c=0 has }$

$\text{equal roots is }\bf\,b^2-4ac\geq\,0$

$\text{Given equation is }\;x^2+mx+2=0$

$\text{Since the equation has real roots, we have}\;b^2-4ac\geq\,0$

$\implies\,m^2-4(1)(2)\geq\,0$

$\implies\,m^2-8\geq\,0$

$\text{Adding 8 on bothsides, we get}$

$\implies\bf\,m^2{\geq}8$

$\therefore\textbf{The required condition is }\;\boxed{\bf\,m^2{\geq}8}$

$\implies\text{Option (C) is correct}$

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Answer 2

The required "option (c) $m^{2}$ ≥ 8" is correct.

Step-by-step explanation:

The given quadratic equation is:

$x^2$ + mx + 2 = 0

Here, a = 1, b = m and c = 2

∴ Discriminant, D = $b^{2}$ - 4ac

= $m^{2}$ - 4(1)(2)

= $m^{2}$ - 8

Given by question,

The roots of the quadratic equation are real roots.

∴ $m^{2}$ - 8 ≥ 0

⇒ $m^{2}$ ≥ 8

Thus, the required "option (c) $m^{2}$ ≥ 8" is correct.