Question

determine the nature of roots √2x^2+4x+2√2=0​

Answer 1

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

Given quadratic equation is

$\rm :\longmapsto\: \sqrt{2} {x}^{2} + 4x + 2 \sqrt{2} = 0$

Concept Used :-

Nature of roots :-

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

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

So, Now

On comparing the given equation with ax² + bx + c = 0, we have

$\red{\rm :\longmapsto\:a = \sqrt{2} \: }$

$\red{\rm :\longmapsto\:b = 4 \: }$

$\red{\rm :\longmapsto\:c = 2 \sqrt{2} \: }$

So, Discriminant is evaluated as

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

On substituting the values of a, b and c, we get

$\rm :\longmapsto\:D = {4}^{2} - 4( \sqrt{2})(2 \sqrt{2})$

$\rm :\longmapsto\:D = 16 - 16$

$\bf\implies \:D = 0$

$\bf\implies \:Equation \: has \: real \: and \: equal \: roots.$

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Additional Information :-

The solution of ax² + bx + c = 0, using quadratic formula is given by

$\: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \tt{ \: x = \frac{ - b \: \pm \: \sqrt{ {b}^{2} - 4ac } }{2a} \: }}$

If Discriminant, D > 0, is a perfect square, then roots of the equation are real and unequal and rational.

If Discriminant, D > 0, is not a perfect square, then roots of the equation are real and unequal and irrational.