The zeroes of the quadratic polynomial x2 + kx + k where k≠0,
A. Cannot both be positive.
B. Cannot both be negative.
C. Are always unequal
D. Are always equal
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.
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
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.
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