What is the value of k? (1) x^2 - px + q = (x - k) (x - r)x2−px+q=(x−k

This is a good example of an abstract quadratics problem, the type that tends to show up about once per exam if you're doing pretty well. It trades on your comfort with the algebra behind quadratics questions.

From a DS perspective, we can narrow the field of answers down to (C) and (E) pretty quickly. It's asking for the (numerical) value of k, and while statement (1) mentions k, it provides no numbers whatsoever. We could, if we multiplied out the right side, get relationships among k, r, p, and q; however, I'd reserve such math for later pending a first look at statement (2), as we might be able to avoid it entirely. In any case, statement (1) alone is insufficient.

Unlike statement (1), statement (2) does provide a concrete number; however, it provides no information about k whatsoever. So statement (2) alone is insufficient as well.

This brings us, as promised, to (C) vs. (E). Because we know from statement (1) that we can find an algebraic relationships among k, r, p, and q, and because statement (2) puts a number to the relationship between r and p, it's a decent idea to FOIL out the right side of the equation in statement (1) and specifically examine the relationship between k, r, and p.

Because of the parallels between the two sides of the equations (and assuming that k, r, p, and q are constants, which the GMAT will usually mention), we can infer that \(r + k = p\) and in turn that \(r - p = -k\). Since \(r - p = 5\) as well, we can say that \(5 = -k\) and that \(-5 = k\). Thus, both statements combined are sufficient, and the answer is (C).