Question

If ` a, b, c in R ` and the quadratic equation ` x^2 + (a + b) x + c = 0 ` has no real roots then
A. `c(a+b+c) gt 0`
B. `c+(a+b+c)c gt 0`
C. `c-c(a+b+c) gt 0`
D. `c(a+b-c) gt 0`

Answer 1

Correct Answer - B
We have, `x^(2) + (a+b) x + c = 0" "...(i)`
Let `f(x) = x^(2) + (a+b) x + c`
Since coefficient of `x^(2)` is positive. So, the equation (i) will not have real roots, if `f(x) gt 0` for all x `in` R.
`rArr" "f(0) f(1) gt 0 and f(0) f(-1) gt 0`
`rArr" "c(1+a+b+c) gt 0 and c(1-a-b+c) gt 0`
`rArr" "c+c(a+b+c) gt 0 and c-c(a+b-c) gt 0`
Hence, option (b) is are true.