Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.
- $\displaystyle x^{2}-Gx+A^{2}=0.$
- $\displaystyle x^{2}-Ax+G^{2}=0.$
- $\displaystyle x^{2}-2Gx+A^{2}=0.$
- $\displaystyle x^{2}-2Ax+G^{2}=0.$
Reveal answer
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Correct answer
Explanation
If roots are r1, r2, then AM = (r1+r2)/2 = A, so r1+r2 = 2A. GM = sqrt(r1*r2) = G, so r1*r2 = G^2. The quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0, which is x^2 - 2Ax + G^2 = 0.
AI explanation
If the arithmetic mean of the roots is A and the geometric mean is G, then the sum of the roots is 2A and the product of the roots is G^2. For any quadratic equation x^2 - (sum of roots)x + (product of roots) = 0, we substitute these values directly into the formula. This yields the required quadratic equation x^2 - 2Ax + G^2 = 0.