Find the nature of roots of the equations:- 2 y 2 - 2 6 y + 3 = 0

Given: 2 y 2 - 2 6 y + 3 = 0 D = b 2 - 4 ac discriminates (differentiates) between the nature of roots. So it is called discriminant. So, on comparing 2 y 2 - 2 6 y + 3 = 0 with the standard form, ax 2 + bx + c = 0 of quadratic equation we find that: a = 2 , b = - 2 6 , c = 3 On putting values in discriminant D = b 2 - 4 ac D = ( - 2 6 ) 2 - 4 ( 2 ) ( 3 ) D = 24 - 24 = 0 Since, the discriminant of given equation is zero, thus both roots of this equation are real and equal.

Find the value of k in 3 x 2 - 2 8 x + k = 0 given that the roots of the quadratic equation are real and equal.

On comparing the quadratic equation 3 x 2 - 2 8 x + k = 0 with the standard from ax 2 + bx + c = 0 , we find that a = 3 , b = - 2 8 , c = k Substituting the above values in D = b 2 - 4 ac D = b 2 - 4 ac D = - 2 8 2 - 4 ( 3 ) ( k ) D = 4 × 8 - 12 k D = 32 - 12 k When the roots are real and equal, D = 0 0 = 32 - 12 k 12 k = 32 k = 32 12 k = 8 3 Hence, the value of k = 8 3 .