Solution:

Introduction:
In this problem, we are given a quadratic equation ax² + bx + c = 0 and we need to find the value of A²/ac bc/a². We are also given that the sum of the roots of the quadratic equation is equal to the sum of the squares of their reciprocals.

Sum of the roots of the quadratic equation:
The sum of the roots of a quadratic equation ax² + bx + c = 0 is given by the formula -b/a. Therefore, the sum of the roots of the given quadratic equation is -b/a.

Sum of the squares of the reciprocals of the roots:
Let the roots of the quadratic equation ax² + bx + c = 0 be α and β. Then, the sum of the squares of the reciprocals of the roots is given by (1/α)² + (1/β)². This can be simplified as follows:

(1/α)² + (1/β)² = (α² + β²)/(αβ)²

Using the quadratic formula, we can express α and β in terms of a, b, and c as follows:

α = (-b + √(b² - 4ac))/2a
β = (-b - √(b² - 4ac))/2a

Substituting these values in the above expression, we get:

(1/α)² + (1/β)² = (α² + β²)/(αβ)²
= [(b² - 4ac)/4a²]/(c/a)²
= (b² - 4ac)/4a²c

Therefore, the sum of the squares of the reciprocals of the roots is (b² - 4ac)/4a²c.

Relationship between the given values:
We are given that the sum of the roots of the quadratic equation is equal to the sum of the squares of their reciprocals. Therefore, we have the equation:

-b/a = (b² - 4ac)/4a²c

Multiplying both sides by -4a³c, we get:

4b²c - 16abc² = -4a³c²

Dividing both sides by -4ac, we get:

a²/ac - bc/a = 4b²c/4a³c

Simplifying, we get:

A²/ac - bc/a² = b²/ac

Therefore, the value of A²/ac bc/a² is b²/ac.