Directions: Read the following quadratic equations carefully and answer the…

Concept

For a set of integers, the L.C.M. (Least Common Multiple) is the smallest positive integer divisible by every member of the set. When the numbers are pairwise coprime (share no common factor other than 1), the L.C.M. equals their product; otherwise each prime factor is taken to its highest power across the set.

A surd √(k·x2) denotes the principal (non-negative) square root, so √(4x2) = 2|x|, √(81y2) = 9|y| and √(625z6) = 25|z3|. In this standard IBPS quadratic format the variables are taken positive, so each radical is read as the positive quantity (2x, 9y, 25z3), which is what makes the equations clean quadratics.

Application

Solve each equation and keep the larger root:

1. Equation (i): x·x − 3x − √(4x2) = −6 gives x2 − 3x − 2x = −6, i.e. x2 − 5x + 6 = 0 → (x − 2)(x − 3) = 0, so x = 2 or 3. Larger root = 3.

2. Equation (ii): y2 − √(81y2) = −20 gives y2 − 9y + 20 = 0 → (y − 4)(y − 5) = 0, so y = 4 or 5. Larger root = 5.

3. Equation (iii): √(625z6) = 25z3, so (z2·25z3)/(5z3) = 5z2. Then 5z2 + 28 = 39z → 5z2 − 39z + 28 = 0. Discriminant = 392 − 4·5·28 = 1521 − 560 = 961 = 312, so z = (39 ± 31)/10 = 7 or 0.8. Larger root = 7.

4. Equation (iv): p2 − 15p = −56 gives p2 − 15p + 56 = 0 → (p − 7)(p − 8) = 0, so p = 7 or 8. Larger root = 8.

The larger roots are 3, 5, 7 and 8. These are pairwise coprime, so L.C.M. = 3 × 5 × 7 × 8 = 840.

Cross-check

Prime factorisation: 3 = 3, 5 = 5, 7 = 7, 8 = 23. The L.C.M. takes 23 · 3 · 5 · 7 = 8 · 105 = 840, and 840 is divisible by each of 3, 5, 7 and 8, confirming the result.