All the math above is right, but I'm afraid the thought process might be wrong.
If what you're looking for is the beta of A with respect to X, the most important thing to understand is that the definition of beta is the OLE regression coefficient of A to X, not the correlation coefficient. The mathematical definition of beta is cov(A,X) / var(X), not corr(A,X) which would be cov(A,X)/ [sd(A) * sd(X)]. Beta is meant to illustrate the sensitivity of A to moves in X (not ignoring scale), so if significantly greater than 1 it's intended to mean "if X moves if one direction, then A will move a lot more in the same direction," if less than 1 (but greater than 0) it means same direction but less, and if negative it means opposite direction with scale depending on the magnitude of the negative number. The correlation coefficient (by definition) will always be between -1 and 1.
To compute the correlation, though, all the math above again is correct. To decompose what's specifically going, the following rules of probability are involved--
1) Cov(A + B + C, A) = Cov(A, A) + Cov(B, A) + Cov(C, A)
2) Cov(bA) = b * Cov(A)
3) Cov(A, A) = Var(A)
4) Var(xA + yB + zC) = x^2 * Var(A) + y^2 * Var(B) + z^2 * Var(C) + x*y*Cov(A, B) + y*z*Cov(B, C) + x*z*Cov(A, C)
5) Corr(A,X) = Cov(A,X) / [sd(A) * sd(X)]
Therefore if what you're specifically given is X = 0.25 * A + 0.25 * B + 0.5 * C, then--
Cov(A, X) = Cov(A, 0.25 *A + 0.25 * B + 0.5 * C) [definition of X]
= Cov(A, 0.25 * A) + Cov(A, 0.25 * B) + Cov(A, 0.25 * C) [rule #1]
= 0.25 * Cov(A, A) + 0.25*Cov(A, B) + 0.25 * Cov(A, C) [rule #2]
= 0.25 * Var(A) + 0.25 * Cov(A, B) + 0.25 * Cov(A, C) [rule #3]
And, Corr(A, X) would be--
Cov(A, X) / [sd(A) * sd(X)] [rule #5]
If what you're looking for is the beta of A with respect to X, though, then you would also need Var(X), which would be--
Var(X) = Var(0.25 * A + 0.25 * B + 0.5 * C) [definition of X]
= 0.25^2*Var(A) + 0.25^2*Var(B) + 0.5^2*Var(C) + 0.25*0.25*Cov(A, B) + 0.25*0.5*Cov(B, C) + 0.25*0.5*Cov(A, C) [rule #4]
And the beta would then be Cov(A, X) / Var(X).
In industry, though, what you'll almost always be looking at in terms of beta is beta of something relative to the market. The first practical question would be how to define "the market," and it almost always depends on the specific stock. Whatever it is, though, it will usually be some form of market index, and the returns of the index will almost always be directly observable... the specific components of the index, the weightings of each, and all the covariances between them would just be extraneous information for doing the calculation. As an example, if you're looking for the "beta of AAPL," it will almost always be computed relative to the S&P500, the returns of the S&P500 can be directly observed, and all you would need to do in practice to compute the beta would be to dump the returns series of AAPL and SPX, compute COVARIANCE.S(AAPL, SPX) / VAR.S(SPX), and call it a day... Whether "the market" should always be defined as the S&P500, though, again would depend on the specific stock... if you were looking at a small-cap US stock for a fund benchmarked to the Russell 2000 or a European stock for something benchmarked to MSCI EAFE, computing beta relative to a large-cap US index would probably be completely useless for what you're trying to do.
As an additional wrinkle, once you land on a consistent definition of "the market," it can still end up being more of an art than a science to determine what returns to even use in the calculation. If you were to look up AAPL on Yahoo Finance right now it would tell you that the beta is ~1.5 or so, but if you were to look it up on a Bloomberg terminal it would tell you something like ~1.2... the reason for this is that while both are computed relative to the S&P, one uses the last 3 years of monthly returns while the other one uses the last 2 years of weekly, and it can often result in two completely different numbers... so be careful.
