If 𝛼 and 𝛽 are the roots of the quadratic equation 𝑎𝑥² + 𝑏𝑥 + 8 = 0, where, 𝑎,𝑏 ∈ 𝑅 and 𝑎 ≠ 0, 𝛼 + 1 and 𝛽 + 1 are the roots of the quadratic equation 𝑎𝑥²﹣52𝑥 + 𝑐 = 0 where 𝑐 ∈ R and, 𝛼﹣1 and 𝛽﹣1 are the roots of the quadratic equation 𝑎𝑥² + 8𝑥 + 𝑑 = 0 where 𝑑 ∈ 𝑅, then find the value of 𝑐 + 𝑑.

By Vieta's formulas, the sum of the roots of ax^2 + bx + 8 = 0 is alpha + beta = -b/a. For ax^2 - 52x + c = 0, the sum of the roots (alpha + 1) + (beta + 1) = 52/a, which means alpha + beta + 2 = 52/a. For ax^2 + 8x + d = 0, the sum of the roots (alpha - 1) + (beta - 1) = -8/a, meaning alpha + beta - 2 = -8/a. Solving these equations gives alpha + beta = 22/a, so c = 8 * 24/a and d = 8 * 20/a, leading to c + d = (192 + 160)/a = 46.

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