Let `x^(2) - ( m - 3) x + m = 0 ( m in` R) be a quadratic equation. Find the value of m for which the roots are
(i) real and distinct
(ii) equal
(iii) not real

Explanation

Let `f(x) = x^(2) - m ( m - 3) x + m = 0`
(i) Both the roots are real and distinct. So, `D gt`0
`rArr (m - 3)^(2) - 4m gt ` 0
or `m^(2) - 10m + 9 gt`0
or `(m -1) (m - 9) gt` 0
`rArr m in (-infty, 1 ) cup (9, infty)`

(ii) Both the roots are equal. So
`D= 0`
`rArr m = 9 or m = 1`

(iii) Both the roots are imaginary. So,
`D lt`0
`rArr ( m - 1) (m - 9) lt` 0
`rArr m in` ( 1, 9)

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