Find the nature of the roots of the following quadratic equations. If the real roots exist, find them

2*x*^{2} - 6*x* + 3 = 0

Determine the nature of the roots of the following quadratic equation:

2*x*^{2} - 6*x* + 3 = 0

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#### Solution

2x^{2} - 6x + 3 = 0

Comparing this equation with ax^{2} + bx + c = 0, we get

a = 2, b = -6, c = 3

Discriminant = b2 - 4ac

= (-6)^{2} - 4 (2) (3)

= 36 - 24 = 12

As b2 - 4ac > 0,

Therefore, distinct real roots exist for this equation

`x = (-b+-b^2-4ac)/(2a)`

`=(-(-6)+-sqrt((-6)^2-4(2)(3)))/(2(2))`

= `(6+-sqrt12)/4=(6+-2sqrt3)/4`

= `(3+-sqrt3)/2`

Therefore the root are `(3+-sqrt3)/2 `

Concept: Nature of Roots

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