The zeros of the quadratic polynomial $x^2 + kx + k, k \neq 0$
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cannot both be positive
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cannot both be negative
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are always unequal
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are always equal
For x^2 + kx + k = 0, product of roots = k and sum of roots = -k. If roots are positive, product must be positive (k > 0) and sum must be positive (-k > 0 => k < 0). This is a contradiction. Thus, they cannot both be positive.
For a quadratic equation x squared + kx + k equals 0, the sum of the roots is negative k and the product of the roots is k. Since k is not equal to zero, both the sum and product of the roots must have opposite signs. This means that if the roots are real, they must have opposite signs, making it impossible for both roots to be positive or both to be negative. Therefore, the zeros cannot both be positive.