We are given an equation of the form ax^2 + bx + c = 0 and we are asked to find the equation whose roots are a + 1/b and b + 1/a.

To solve this problem, we will first find the sum and product of the roots of the given equation. Then, using these values, we will construct the new equation.



Let's denote the roots of the given equation ax^2 + bx + c = 0 as a and b.

The sum of the roots can be found using the formula: sum of roots = -b/a.
So, the sum of the roots is a + b = -b/a.

The product of the roots can be found using the formula: product of roots = c/a.
So, the product of the roots is ab = c/a.



Now, we need to find the roots of the new equation. Let's denote the roots of the new equation as x1 and x2.

We are given that the roots of the new equation are a + 1/b and b + 1/a.

Using the sum of roots formula, we can write:
x1 + x2 = (a + 1/b) + (b + 1/a)

Simplifying this expression, we get:
x1 + x2 = a + b + 1/b + 1/a

Using the product of roots formula, we can write:
x1 * x2 = (a + 1/b) * (b + 1/a)

Simplifying this expression, we get:
x1 * x2 = ab + a/b + b/a + 1

Now, we can construct the new equation using the sum and product of the roots:
(x - x1) * (x - x2) = 0

Substituting the values of x1 + x2 and x1 * x2, we get:
(x - (a + b + 1/b + 1/a)) * (x - (ab + a/b + b/a + 1)) = 0

Simplifying further, we get the final equation:
x^2 - (a + b + 1/b + 1/a)x + (ab + a/b + b/a + 1) = 0

Therefore, the equation whose roots are a + 1/b and b + 1/a is:
x^2 - (a + b + 1/b + 1/a)x + (ab + a/b + b/a + 1) = 0.