Two numbers differ by 12. The square of the larger exceeds twelve times the smaller by a quantity of Z. Find the minimum value of Z.

Answer 1

L-S = 12

L = 12+S

L^2 = 12S + Z

(12+S)^2 = 12S + Z

144 +24S + S^2 = 12S + Z

S^2 + 12S + 144 = Z

S^2 + 12S + 36 +144-36 = Z

(S+6)^2 = Z-108

S+6 = + or - sqr(Z-108)

S = 6 + sqr(Z-108)

L = 18 + sqr(Z-108)

Z>108. 108 is the minimum value for Z

otherwise the numbers L and S would be imaginary