If f(x) = ax² + bx + c has no real zeroes and a + b + c < 0, then find the nature of f(x)

Video Explanation

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Given

f(x) = ax² + bx + c

The polynomial has no real zeroes.

a + b + c < 0

To Find

The nature (sign) of the polynomial f(x).

Solution

Step 1: Use the Condition of No Real Zeroes

If a quadratic polynomial has no real zeroes, then:

b² − 4ac < 0

This means the graph of the polynomial does not intersect the x-axis.

Therefore, the polynomial is either:

• always positive, or
• always negative

Step 2: Use the Given Condition a + b + c < 0

We know that:

f(1) = a + b + c

Given that:

f(1) < 0

So the value of the polynomial at x = 1 is negative.

Step 3: Draw the Conclusion

Since:

• f(x) has no real zeroes (does not cross x-axis), and
• f(1) is negative

The polynomial must be negative for all real values of x.

Hence, the leading coefficient a must be negative.

Final Answer

The polynomial f(x) is negative for all real values of x.

Conclusion

Thus, if the quadratic polynomial f(x) = ax² + bx + c has no real zeroes and a + b + c < 0, then f(x) < 0 for all real values of x.