Let `x^2-(m-3)x+m=0 (mepsilonR)` be a quadratic equation . Find the values of m for which the roots are real and distinct

Explanation

Here `a=1,b=2(m-1)` and `c=m+5`
`:.D=b^(2)-4ac=4(m-1)^(2)-4(m+5)`
`=4(m^(2)-3m-4)`
`:.D=4(m-4)(m+1)` and here `a=1 ge0`
(i) `b=0` and `D gt0`
`implies2(m-1)=0` and `4(m-4)(m+1)gt0`
`impliesm=1` and `m epsilon (-oo, 1)uu(4,oo)`
`:.m epsilon phi` [null set]
(ii) `a=c` and `Dge0`
`implies 1=m+5` and `4(m-4)(m+1)ge0`
`implies m=-4` and `m epsilon(-oo,-1]uu[4,oo)`
`:.m=-4`
(iii) `agt0, clt0` and `Dgt0`
`implies1`gt0,m+5lt0` and `4(m-4)(m+1)gt0`
`impliesmlt-5` and `m epsilon (-oo,-1)uu(4,oo)`
`:.m epsilon(-oo,-5)`
(iv) `agt0, blt0, cgt0` and `Dge0`
`implies1 gt0,2(m-1)lt0,m+5gt0`
and `4(m-4)(m+1)ge0`
`impliesmlt1,mgt-5` and `m epsilon (-oo,-1]uu[4,oo)`
`impliesm epsilon (-5,-1]`
(v) `agt0, bgt0,cgt0` and `Dge0`
`implies1gt0,2(m-1)gt0,m+5gt0`
and `4(m-4)(m+1)ge0` ltbr. `impliesmgt1,mgt=5` and `m epsilon (-oo,-1]uu[4,00O`
`:.m epsilon[4,oo)`
(vi) Either one root is positive or both roots are positive
i.e (c) `uu` (d)
`implies m epsilon (-oo,-5)uu(-5,-1]`
(vii) Either one root is negative or both roots are negative
i.e. (c) `uu` (e)
`impliesm epsilon (-oo,-5)uu[4,oo)`