Find the nature of the roots of the equations of given below :
(a) `x^(2) - 13x + 11 =0`
(b ) `18x^(2) - 14x + 17 = 0`
(c ) `9x^(2) - 36x + 36 = 0`
(d) `3x^(2) - 5x - 8 = 0`

Explanation

(a) Given `x^(2) - 13x + 11 = 0`
Comparing it with `ax^(2) + bx + c = 0`, we have `a = 1, b = -13` and `c = 11`.
Now, `b^(2) - 4ac = (-13)^(2) - 4(1)(11)`
`= 169 - 44 = 125 lt 0`.
`rArr b^(2) - 4ac gt 0` and is not perfet square.
`:.` The roots are distinct and irrational .
(b) Given `18x^(2) - 14x + 17 =0`
Comparing it with `ax^(2) + bx + c = 0, a = 18, b = -14` and c = 17.
Now , `b^(2) - 4ac = (-14)^(2) - 4(18) (17)`
`= 196 - 1224`
`= - 1028 lt 0`
`rArr b^(2)-4ac lt 0`.
`:.` The roots are imagniary.
(c) Given `9x^(2) - 36x + 36 = 0`.
`rArr 9(x^(2) - 4x+4) =0`
`rArr x^(2) - 4x + 4 = 0`
Comparing the above equation , with `ax^(2) + bx + c= 0, a= 1, b = - 4` and c = 4
Now, `b^(2)- 4ac = (-4)^(2) - 4(1)(4)`
`= 16 - 16 = 0`
`rArr b^(2) - 4ac = 0`
`:.` The roots are equal.
(d) Given `3x^(2) - 5x - 8 = 0`
Comparing the above equation with `ax^(2) + bx + c = 0`, we getm `a = 3, b = -5` and `c = - 8`.
Now, `b^(2)- 4ac = (-5)^(2) - 4(3) (-8)`
`= 25+96 = 121 gt 0` ,
`rArr b^(2) - 4ac lt 0` and is a perfect square.
`:.` The roots are rational and distinct.