If x is added to each of 14, 12, 34 and 30, the numbers so obtained, in this order, are in proportion. What is the value of √(12x + 9)?

If numbers are in proportion: (14+x)/(12+x) = (34+x)/(30+x). Cross multiplying: (14+x)(30+x) = (12+x)(34+x) → 420 + 14x + 30x + x² = 408 + 12x + 34x + x² → 420 + 44x = 408 + 46x → 12 = 2x → x = 6. So √(12×6+9) = √81 = 9.