For the roots of the equation $a-bx-x^2=0$; $(a > 0; b > 0)$ which statement is true-
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positive and same sign
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negative and same sign
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greater root in magnitude, negative and opposite in signs
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greater root in magnitude, positive and opposite in signs
The equation is x^2 + bx - a = 0. The product of roots is -a. Since a > 0, the product is negative, meaning the roots have opposite signs. The sum of roots is -b. Since b > 0, the sum is negative, meaning the root with the larger absolute value must be negative.
The roots of the quadratic equation -x^2-bx+a=0 are found using the quadratic formula. The roots are [-b + sqrt(b^2+4a)]/(-2) and [-b - sqrt(b^2+4a)]/(-2), which simplify to [b - sqrt(b^2+4a)]/2 and [b + sqrt(b^2+4a)]/2. Since a and b are both positive, the square root term is strictly greater than b, making the first root positive and the second root negative. Comparing their magnitudes, the negative root has a larger absolute value because its numerator is a sum of two positive numbers. The roots have opposite signs and the greater root in magnitude is negative.