Question

If `c` is positive and `2a x^2+3b x+5c=0` does not have aby real roots, then prove that `2a-3b+5b<0.`

Answer 1

Given ` c ge 0 and 2ax^(2) + 3bx + 5c = 0 ` does not have real root .Let
`f(x) = 2ax^(2) + 3 bx + 5c`
`rArr f(x) gt 0, AA x in` R, if a `gt 0 or f(x)lt 0 Aax in` R, if a `lt` o
But `5c = f(0) gt 0` ltbr gt `rArr f(x) gt 0, AA z in` R
`rArr 2ax^(2) + 3bx + 5c gt 0, AA x in` R
`rArr 2a - 3b + 5c gt` 0 (for x = - 1) .