Question

It is given that the equation (mx)²+ 5nx+4=0 has two equal real roots while the quadratic equation px²-x+m=0 has two distinct roots where m,n and p are constants. Express the range of n in terms of p

Answer 1

Given:

`(mx)^2+5nx+4=0\text{ has two equal real roots}`
`px^2-x+m=0\text{ has distict roots. }`

Taking the first equation:

`a=m^2\text{ ; b=5n ; c=4 }`
`b^2-4ac=0`
`25n^2-4m^2(4)=0`
`(5n)^2-(4m)^2=0`
`(5n+4m)(5n-4m)=0`
`5n+4m=0\text{ ; 5n-4m=0}`
`m=-(5)/(4)n\text{ ; m=}(5)/(4)n`
`4m=-5n\text{ ; 4m=5n}`

Taking the second equation:

`a=p\text{ ; b=-1 ; c=m}`
`b^2-4ac>0`
`1-4pm>0`

If 4m=5n,

`1-p(5n)>0`
`1>p(5n)`
`(1)/(5p)>n`

If 4m=-5n,

`1-p(-5n)>0`
`1+5pn>0`
`5pn>-1`
`n>-(1)/(5p)`

Range of n in terms of p:

`(1)/(5p)>n>-(1)/(5p)`[tex]-\frac{1}{5p}