Find the number of integer quadruples (a,b,c,d) with 0\leq a,b,c,d \leq 100, such that a and b are the roots of the quadratic equation x^2-cx+d=0, while c and d are the roots of the quadratic equation x^2-ax+b.
To find the number of integer quadruples (a, b, c, d) with the given conditions, we need to first determine the relationships between a, b, c, and d based on the given quadratic equations.
Let's start with the first equation: x^2 - cx + d = 0. We know that a and b are the roots of this equation.
By Vieta's formulas, the sum of the roots of this quadratic equation is equal to the negation of the coefficient of the linear term (c), and the product of the roots is equal to the constant term (d).
So we have the following relationships:
a + b = c (Equation 1)
ab = d (Equation 2)
Now let's move on to the second quadratic equation: x^2 - ax + b = 0. We know that c and d are the roots of this equation.
Again, by Vieta's formulas, the sum of the roots is equal to the coefficient of the linear term (a), and the product of the roots is equal to the constant term (b).
So we have the following relationships:
c + d = a (Equation 3)
cd = b (Equation 4)
Now that we have established these relationships, we can use them to find the number of integer quadruples (a, b, c, d) that satisfy these conditions.
We can start by finding the values of a and b in terms of c and d using Equation 1 and Equation 2:
From Equation 1, we have a + b = c.
Substituting b = c - a into Equation 2, we get a(c - a) = d.
Rearranging, we have: a^2 - ac + d = 0.
Now let's rewrite the equations to solve for the values of c and d in terms of a and b:
From Equation 3, we have c + d = a.
Substituting d = a - c into Equation 4, we get ac - a^2 + b = 0.
Rearranging, we have: a^2 - ac + b = 0.
Now we have two quadratic equations with a common discriminant (a^2 - ac + b = 0 and a^2 - ac + d = 0), so we can equate their discriminants and solve for a, b, c, and d.
(a^2 - ac + b)^2 - 4(a^2)(b) = (a^2 - ac + d)^2 - 4(a^2)(d).
Expanding the squares and simplifying, we get:
a^4 - 2a^3c + a^2(c^2 - 3b + 2d) + 2acd - 4ab + 4bd = a^4 - 2a^3c + a^2(c^2 - 2cd + d^2) - 4a^2d.
Simplifying further, we have:
a^2(c^2 - 3b + 2d) + 2acd - 4ab + 4bd = a^2(c^2 - 2cd + d^2) - 4a^2d.
Canceling out the common terms, the remaining equation is:
2acd - 4ab + 4bd = -4a^2d.
Now we can analyze this equation to determine the number of integer quadruples (a, b, c, d) that satisfy the given conditions. However, this equation is quite complex to solve algebraically due to its non-linearity and the number of variables involved. Therefore, it would be more efficient to use a computer program or algorithm to iterate through the possible values of a, b, c, and d in the given range (0 ≤ a, b, c, d ≤ 100), and count the number of valid solutions.
By evaluating this equation for all possible integer values of a, b, c, and d in the specified range, you can determine the exact number of integer quadruples that satisfy the given conditions.