Let `x^2-(m-3)x+m=0 (mepsilonR)` be a quadratic equation . Find the values of m for which the roots are (i) real and distinct (ii) equal (iii) not real (iv) opposite in sign (v) equal in magnitude but opposite in sign (vi) positive (vii)negative (viii) such that atleast one is positive

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)`