What is the sum of all possible positive integer values of n, such that n^2+19n+130 is a perfect square?
To find the sum of all possible positive integer values of n, for which the expression \(n^2 + 19n + 130\) is a perfect square, we can follow the steps below:
Step 1: Express the given equation as a quadratic equation: \(n^2 + 19n + 130 = k^2\), where k is a positive integer representing the perfect square.
Step 2: Rearrange the equation: \(n^2 + 19n + (130 - k^2) = 0\).
Step 3: Apply the quadratic formula to find the solutions for n.
The quadratic formula is given by: \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
In our case, a = 1, b = 19, and c = 130 - k^2.
Step 4: Substitute the values of a, b, and c into the quadratic formula: \(n = \frac{-19 \pm \sqrt{19^2 - 4(1)(130 - k^2)}}{2(1)}\).
Step 5: Simplify the equation: \(n = \frac{-19 \pm \sqrt{361 - 520 + 4k^2}}{2}\).
Step 6: Further simplify: \(n = \frac{-19 \pm \sqrt{4k^2 - 159}}{2}\).
Since n should be a positive integer, the discriminant (\(4k^2 - 159\)) must be a perfect square.
Step 7: Find the perfect squares less than or equal to 159 (since we don't need to consider perfect squares greater than 159).
The perfect squares less than or equal to 159 are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and 121.
Step 8: For each perfect square, solve for k: \(4k^2 - 159 = perfect \ square\).
For example, when the perfect square is 0, we have \(4k^2 - 159 = 0\), which yields the only solution: k = 0.
Step 9: Substitute the value of k back into the simplified equation to find n.
For example, for k = 0, we have \(n = \frac{-19 \pm \sqrt{4(0)^2 - 159}}{2}\), which simplifies to n = -19/2 or n = 0.
Repeat this step for all perfect squares found in Step 7.
Step 10: Add up all the positive integer values of n obtained in Step 9.
For example, if we found the values n = 0 and n = 5 for the perfect square 9, we would add them together: 0 + 5 = 5.
Continue this process for all perfect squares found in Step 7.
Step 11: Calculate the sum of all the positive integer values of n obtained in Step 10.
This sum will give you the answer to the original question: the sum of all possible positive integer values of n such that \(n^2 + 19n + 130\) is a perfect square.