Given that positive numbers a, B and C satisfy A2 + C2 = 16 and B2 + C2 = 25, the value range of k = A2 + B2 is______ ．

Given that positive numbers a, B and C satisfy A2 + C2 = 16 and B2 + C2 = 25, the value range of k = A2 + B2 is______ ．

∵ positive numbers a, B, C satisfy A2 + C2 = 16, B2 + C2 = 25, ∵ C2 = 16-a2, A2 > 0, so 0 < C2 < 16. Similarly, C2 = 25-b2 gets 0 < C2 < 25, so 0 < C2 < 16. Two formulas are added: A2 + B2 + 2c2 = 41, that is A2 + B2 = 41-2c2, and ∵ - 16 < - C2 < 0, that is - 32 < - 2c2 < 0 ∵ 9 < 41-2c2 < 41, that is 9 < K < 4

Given that the real numbers a, B, C satisfy a + B + C = 0, ABC = 16. Find the value range of C
I think when C > 0, C should be ≥ 4. If C < 0, C should be ≤ 4

A + B = - C, ab = 1 / 6C, then - C and 16 / C are the two roots of the equation x & # 178; + CX + 16 / C = 0. The discriminant of this equation is C & # 178; - 64 / C ≥ 0, and the solution is C ≥ 4 or C ≤ 0