Key Concepts and Formulas

• Quadratic Function (Vertex Form): $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex. The given function $f(x) = (x-p)^2 - q$ is in this form with $a=1$, vertex at $(p, -q)$.
• Roots of a Quadratic Equation: For $f(x) = 0$, the roots are the values of $x$ that satisfy the equation. For $f(x) = (x-p)^2 - q = 0$, the roots are $x = p \pm \sqrt{q}$.
• Absolute Difference Between Roots: For roots $r_1, r_2$, this is $|r_1 - r_2|$. For $f(x) = (x-p)^2 - q = 0$, the absolute difference is $|(p+\sqrt{q}) - (p-\sqrt{q})| = |2\sqrt{q}| = 2\sqrt{q}$ (since $q>0$).
• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant (the common difference, $D$). If the mean of $n$ terms is $p$, and $n$ is even, the terms can be symmetrically represented as $p \pm \frac{D}{2}, p \pm \frac{3D}{2}, \dots$.
• Properties of Absolute Value: If $|A| = |B|$, then either $A=B$ or $A=-B$.

Step-by-Step Solution

Step 1: Represent the Arithmetic Progression Terms We are given that $a_1, a_2, a_3, a_4$ are in an arithmetic progression with mean $p$ and a positive common difference, let's call it $D$. Since there are four terms and their mean is $p$, we can represent them symmetrically around $p$. Let $d = D/2$. Since $D>0$, we have $d>0$. The terms are: $a_1 = p - 3d$ $a_2 = p - d$ $a_3 = p + d$ $a_4 = p + 3d$ Explanation: Representing the terms symmetrically simplifies calculations when substituted into $f(x) = (x-p)^2 - q$, as the $p$ terms will cancel out. The condition $D>0$ implies $d>0$.

Step 2: Evaluate $f(a_i)$ for each term Substitute the AP terms into the function $f(x) = (x-p)^2 - q$: For $a_1 = p - 3d$: $f(a_1) = ((p - 3d) - p)^2 - q = (-3d)^2 - q = 9d^2 - q$ For $a_2 = p - d$: $f(a_2) = ((p - d) - p)^2 - q = (-d)^2 - q = d^2 - q$ For $a_3 = p + d$: $f(a_3) = ((p + d) - p)^2 - q = (d)^2 - q = d^2 - q$ For $a_4 = p + 3d$: $f(a_4) = ((p + 3d) - p)^2 - q = (3d)^2 - q = 9d^2 - q$ Explanation: This step expresses the values of the function at the AP points in terms of $d$ and $q$. We observe that $f(a_1) = f(a_4)$ and $f(a_2) = f(a_3)$, which is expected due to the symmetry of the AP terms around $p$ and the quadratic nature of $f(x)$.

Step 3: Apply the given condition $|f(a_i)| = 500$ We are given that $|f(a_i)| = 500$ for all $i \in \{1, 2, 3, 4\}$. This leads to the following distinct equations:

1. $|9d^2 - q| = 500$
2. $|d^2 - q| = 500$ Explanation: This step translates the problem's condition into a system of equations involving $d$ and $q$.

Step 4: Solve the system of absolute value equations From $|9d^2 - q| = 500$ and $|d^2 - q| = 500$, we must have $|9d^2 - q| = |d^2 - q|$. Using the property that if $|A| = |B|$, then $A=B$ or $A=-B$: Case 1: $9d^2 - q = d^2 - q$ $9d^2 = d^2$ $8d^2 = 0$ $d^2 = 0 \implies d = 0$ Explanation: This case leads to $d=0$. However, from Step 1, we know that $d>0$ (since the common difference $D$ is positive). Thus, this case is invalid.

Case 2: $9d^2 - q = -(d^2 - q)$ $9d^2 - q = -d^2 + q$ $9d^2 + d^2 = q + q$ $10d^2 = 2q$ $5d^2 = q$ Explanation: This case yields a valid relationship between $d^2$ and $q$.

Step 5: Use the valid equation to find $q$ Substitute $q = 5d^2$ into one of the original absolute value equations, for example, $|d^2 - q| = 500$: $|d^2 - (5d^2)| = 500$ $|-4d^2| = 500$ $4d^2 = 500$ (since $d^2 \ge 0$) $d^2 = \frac{500}{4} = 125$ Now, we can find $q$ using $q = 5d^2$: $q = 5 \times 125 = 625$ Explanation: By substituting the relationship found in Case 2 into one of the given conditions, we can solve for $d^2$ and then subsequently for $q$. We are given $q>0$, and $q=625$ satisfies this.

Step 6: Calculate the absolute difference between the roots of $f(x)=0$ The roots of $f(x) = (x-p)^2 - q = 0$ are $x = p \pm \sqrt{q}$. The absolute difference between the roots is $|(p + \sqrt{q}) - (p - \sqrt{q})| = |2\sqrt{q}| = 2\sqrt{q}$ (since $q>0$). Substitute the value of $q$ we found: Absolute difference $= 2\sqrt{625}$ Absolute difference $= 2 \times 25$ Absolute difference $= 50$ Explanation: The problem asks for the absolute difference between the roots of $f(x)=0$. We use the formula for this difference and substitute the value of $q$ we determined.

Common Mistakes & Tips

• Ignoring the condition $d>0$: The case $d=0$ arises from $|A|=|B|$ but is invalid because the common difference of the AP is positive.
• Incorrectly setting up AP terms: Using $p-3d, p-d, p+d, p+3d$ is crucial for symmetry and simplification.
• Forgetting $q>0$: While $q=625$ naturally satisfies this, always check constraints.

Summary

The problem involves a quadratic function and an arithmetic progression. By representing the AP terms symmetrically around the mean $p$, we simplified the function evaluations $f(a_i)$. The condition $|f(a_i)|=500$ led to a system of absolute value equations. Solving this system, we found the value of $q$. Finally, we used the formula for the absolute difference between the roots of a quadratic equation, $2\sqrt{q}$, to find the required answer.

The final answer is $\boxed{50}$.