If ( (x+k)^2=x^2+24x+144 ), what is the value of (k)?

Correct answer: (12)

The direct answer is C, 12. Compare the expression with \\((x+k)^2=x^2+2kx+k^2\\). The coefficient of x on the right is 24, so 2k=24 and k=12. The constant term also confirms this: k^2=144, and the positive value giving 2k=24 is k=12. Substitution checks it: \\((x+12)^2=x^2+24x+144\\). Option A, 6, would give middle coefficient 12 and constant 36. Option B, 10, would give 20x and 100. Option C is correct because it satisfies both the middle and final terms. Option D, 24, would give 48x and 576. A common mistake is to use only the last term and forget to check the middle term; always compare both parts.