Condition for No Real Zeroes of a Quadratic Polynomial

Video Explanation

Question

If the quadratic polynomial

\[ f(x) = ax^2 + bx + c \]

has no real zeroes and

\[ a + b + c < 0, \]

then find the sign of \(a\).

Solution

Step 1: Use the Property of Quadratic Polynomials

If a quadratic polynomial has no real zeroes, then its graph does not intersect the x-axis.

Hence, the polynomial has the same sign for all real values of \(x\).

Step 2: Evaluate the Polynomial at \(x = 1\)

\[ f(1) = a + b + c \]

Given:

\[ a + b + c < 0 \Rightarrow f(1) < 0 \]

Step 3: Deduce the Sign of \(a\)

Since the polynomial has the same sign everywhere and \(f(1) < 0\),

the parabola must open downwards.

Therefore,

\[ a < 0 \]

Conclusion

\[ \boxed{a < 0} \]