Introduction:
In this problem, we are given a quadratic equation and we need to find the values of 'n' for which the roots of the equation are equal integers. To solve this problem, we will use the discriminant of the quadratic equation and apply the conditions for the roots to be equal integers.

Quadratic Equation:
The given quadratic equation is x^2 + x(14-n) - 14n + 1 = 0.

Discriminant:
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.

Conditions for Equal Integer Roots:
For the roots of a quadratic equation to be equal integers, the following conditions must be satisfied:
1. The discriminant D must be a perfect square.
2. The roots must be rational numbers.

Solution:
Let's solve this problem step by step:

Step 1: Write the quadratic equation in standard form.
x^2 + x(14-n) - 14n + 1 = 0

Step 2: Identify the coefficients of the equation.
a = 1, b = 14-n, c = -14n + 1

Step 3: Calculate the discriminant.
D = (14-n)^2 - 4(1)(-14n + 1)
= (n^2 - 28n + 196) + 56n - 4
= n^2 + 28n + 192

Step 4: Check if the discriminant is a perfect square.
To check if D is a perfect square, we can factorize it as D = (n + p)^2, where p is an integer.

Expanding (n + p)^2, we get n^2 + 2np + p^2.

Comparing this with the expression for D, we can equate the corresponding coefficients:
n^2 + 28n + 192 = n^2 + 2np + p^2

By comparing the coefficients, we get:
2np = 28n => p = 14

Step 5: Substitute the value of p in the expression for D.
D = (n + 14)^2

Step 6: Check if the roots are rational numbers.
To check if the roots are rational, we need to find the discriminant of the quadratic equation formed by equating the expression inside the square root sign of the quadratic formula to be a perfect square.

The quadratic formula is given by:
x = (-b ± √D) / 2a

In our case, a = 1, b = 14-n, and c = -14n + 1.

Substituting these values, we get:
x = (-(14-n) ± √[(n + 14)^2]) / 2

Step 7: Simplify the expression inside the square root.
√[(n + 14)^2] = n + 14

Step 8: Substitute the simplified expression in the quadratic formula.
x = (-(14-n) ± (n