If $a,b,c\in R$ and the quadratic equation $ax^{2}+bx+c=0$ has no real roots, then
- $(a + b + c) c > 0 $
- $c < 0$
- $a + b + c > 0$
- $a + b + c < 0$
If a quadratic has no real roots, it is always positive (if a > 0) or always negative (if a < 0). Thus, f(x) = ax^2 + bx + c never changes sign. f(0) = c and f(1) = a + b + c must have the same sign. Therefore, their product c(a + b + c) must be positive.
Let f(x) = ax^2 + bx + c. Since the quadratic equation has no real roots, the graph of f(x) never crosses the x-axis, meaning f(x) always maintains the same sign for all real numbers. Additionally, the sign of f(x) must perfectly match the sign of the leading coefficient a, giving a multiplied by f(x) > 0. Evaluating f(0) yields f(0) = c, which implies that a and c must share the same sign, meaning the product ac > 0. Evaluating the function at x = 1 gives f(1) = a + b + c, and because the sign of f(1) matches the sign of a, it follows that a multiplied by (a + b + c) > 0. Multiplying the two positive quantities (ac) and (a multiplied by (a + b + c)) yields a^2 multiplied by c(a + b + c) > 0. Since a^2 is strictly positive for any real non-zero a, we divide by a^2 to deduce the final result, c(a + b + c) > 0.