Question

The zeroes of the quadratic polynomial x² + kx + k, k? 0,​

Answer 1

Answer:

can't not be positive Product of zeroes = k

The sign is positive it means both the zeroes have same sign.

Sum of zeroes = -k

The sign is negative and both have same sign hence the zeroes are both negative

Answer 2

Answer:

Let p(x) = x2 + kx + k ⇒ k (k - 4) > 0 ⇒ k ∈ (- ∞, 0) ⋂ (4, ∞) Here k lies in two intervals, therefore we need to consider both the intervals separately. Case1: When k (- ∞, 0) i.e. k < 0 we know that in a quadratic equation p(x) = ax2 + bx + c, if either a > 0, c < 0 or a < 0, c > 0, then the zeroes of the polynomial are of opposite signs. Here a = 1 > 0, b = k < 0 and c = k < 0 ⇒ both zeroes are of opposite signs Case2: When k ∈ (4, ∞) i.e. k > 0 We know, in quadratic polynomial if the coefficients of the terms are of the same sign, then the zeroes of the polynomial are negative. i.e. if either a > 0, b > 0 and c > 0 or a < 0, b < 0 and c < 0, then both zeroes are negative Here a = 1 > 0, b = k > 0 and c = k > 0 ⇒ both the zeroes are negative Hence, by both cases, both the zeroes cannot be positive.Read more on Sarthaks.com - https://www.sarthaks.com/877953/the-zeroes-of-the-quadratic-polynomial-x-2-kx-k-where-k-0