$a, b, c\in R, a\neq 0$ and the quadratic equation $ax^{2}+bx+c=0$ has no real roots, then which one of the following is not true?
- $a+b+c>0$
- $a(a+b+c)>0$
- $ac(a+b+c)>0$
- $c(a+b+c)>0$
If ax^2 + bx + c = 0 has no real roots, the parabola never touches the x-axis. Thus, the function f(x) = ax^2 + bx + c is always positive (if a > 0) or always negative (if a < 0). Option A states a+b+c > 0, which is f(1). This is not necessarily true because f(x) could be entirely negative if a < 0.
Because the quadratic equation $ax^2 + bx + c = 0$ has no real roots, the function $f(x) = ax^2 + bx + c$ never crosses the x-axis, meaning it maintains the same sign for all real numbers. Therefore, $f(1) = a + b + c$ must share the same sign as the leading coefficient $a$, making $a(a + b + c) > 0$ a true statement. Similarly, $f(0) = c$ must also share the same sign as $a$, which implies $ac > 0$. Because both $ac$ and $a + b + c$ share the same sign, their product $ac(a + b + c)$ must be strictly positive, making this statement true as well. Consequently, multiplying the positive value $a(a + b + c)$ by the positive value $c$ guarantees that $c(a + b + c) > 0$ is true. By elimination, the statement $a + b + c > 0$ cannot be guaranteed, as the roots could all yield a negative sum, making it the false claim.