Problem:
If the roots of the quadratic equation ax^2 + bx + c = 0 are in the ratio p/q, find the value of (b^2)/(ca).

Solution:
To solve this problem, we will use the properties of quadratic equations and the relationship between the roots and coefficients of a quadratic equation.

Step 1: Relationship between roots and coefficients:
Let the roots of the quadratic equation ax^2 + bx + c = 0 be α and β. According to the given condition, the roots are in the ratio p/q, which can be written as:

α/β = p/q

Step 2: Relationship between sum and product of roots and coefficients:
The sum of the roots (α + β) and the product of the roots (α * β) are related to the coefficients of the quadratic equation as follows:

α + β = -b/a
α * β = c/a

Step 3: Solving for α and β:
Using the relationship between the sum and product of roots, we can express α and β in terms of p, q, a, b, and c:

α + β = -b/a
α * β = c/a

Since α/β = p/q, we can rewrite α and β as:

α = (p/q)β

Substituting this into the equation α + β = -b/a, we get:

(p/q)β + β = -b/a
((p/q) + 1)β = -b/a
β = -b/a * (q/(p + q))

Similarly, substituting α = (p/q)β into the equation α * β = c/a, we get:

(p/q)β * β = c/a
((p/q)β^2) = c/a
β^2 = c/a * (q/p)

Step 4: Finding the value of (b^2)/(ca):
Now, we can substitute the value of β^2 into the expression (b^2)/(ca):

(b^2)/(ca) = (b^2)/(c/a * (q/p))
= (b^2 * p)/(c * a * q)

Therefore, the value of (b^2)/(ca) is (b^2 * p)/(c * a * q).

Summary:
The value of (b^2)/(ca) is (b^2 * p)/(c * a * q). This result is obtained by using the relationship between the roots and coefficients of a quadratic equation and solving for the roots in terms of p, q, a, b, and c.