Consider the inequation `x^(2) + x + a - 9 lt 0`
The values of the real parameter a so that the given inequations has at least one negative solution.

Options

A. `(-oo,9)`
B. `((37)/(4),oo)`
C. `(-oo, (37)/(4))`
D. none of these

Answer

3

Explanation

If `x^(2) + x + a-9 lt 0` has at least one negative solution, then either both the roots of equations `x^(2) + x + a - 9=0` are non-posititve or 0 lies between the roots.
For case I, sum of roots is `a - 9 gt 0`
`rArr a gt 9 and `
`D gt 0 `
`rArr 1-4(a-9) gt 0 rArr a lt (37)/(4)`
Hence, `9 le a le 37//4`.
For case II, `f(0) gt 0`
`rArr a lt 9 rArr a in (-oo,(37)/(4))`