Question

Determine the nature of the roots for the following quadratic equations: 2x²+5√3x+6=0

Answer 1

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm :\longmapsto\: {2x}^{2} + 5 \sqrt{3}x + 6 = 0$

We know,

Nature of roots depends upon the Discriminant of quadratic equation.

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D).

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Here,

$\rm :\longmapsto\:a = 2$

$\rm :\longmapsto\:b = 5 \sqrt{3}$

$\rm :\longmapsto\: c= 6$

So,

$\rm :\longmapsto\:Discriminant, D = {b}^{2} - 4ac$

$\rm \:  =  \:  \: {(5 \sqrt{3}) }^{2} - 4 \times 2 \times 6$

$\rm \:  =  \:  \: 75 - 48$

$\rm \:  =  \:  \: 27$

$\bf\implies \:Discriminant, D = 27 > 0$

Hence,

Roots of the quadratic equation are real and unequal.