If the zeroes of a quadratic polynomial ax^2 + bx + c are both positive, then a, b and c all have the same sign.

Question

Is the following statement True or False? Justify your answer.

If the zeroes of a quadratic polynomial ax² + bx + c are both positive, then a, b and c all have the same sign.

Answer 1

If the zeroes of a quadratic polynomial ax² + bx + c are both positive, then a, b and c all have the same sign: False.

Let α, β be the zeroes of the polynomial p(x) = ax² + bx + c

Sum of the zeroes = - (coefficient of x) ÷ coefficient of x²

α + β = - b/a > 0 (∵ α > 0, β > 0 ⇒ α + β > 0)

∴ for – b/a > 0, b and a must have opposite signs.

Product of the zeroes = constant term ÷ coefficient of x²

αβ = c/a > 0 (∵ α,β > 0⇒ αβ > 0)

∴ for c/a > 0, c and a must have same signs.

Case 1: when a > 0

⇒ - b > 0 and c > 0

= b < 0 and c > 0

Case 2: when a < 0

⇒ - b < 0 and c < 0

= b > 0 and c < 0

Hence, the coefficients have different signs.