Multiple choice

State whether the following statement is true or false. If A and G are respectively arithmetic and geometric means between two positive quantities a and b, then the quadratic equation having a, b as its roots is $x^2-2Ax+G^2=0$.

  1. True

  2. False
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A Correct answer
Explanation

The arithmetic mean of two positive numbers a and b is A = (a + b) / 2, which means their sum is 2A. Their geometric mean is G = sqrt(a * b), which means their product is G^2. A quadratic equation with roots a and b is represented by x^2 - (sum of roots)x + (product of roots) = 0, which simplifies to x^2 - 2Ax + G^2 = 0.

AI explanation

By definition, the arithmetic mean A of two quantities a and b is (a + b) / 2, meaning their sum is 2A. The geometric mean G is the square root of their product, meaning ab = G^2. A quadratic equation with roots a and b takes the form x^2 - (sum)x + (product) = 0, so substituting the values gives x^2 - 2Ax + G^2 = 0, making the statement true.

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