If the roots of the equation x² - px + 54 = 0 are in the ratio 2 : 3, then the…

Concept: For a monic quadratic x² − Sx + P = 0, Vieta's formulas give sum of roots = S and product of roots = P. When two roots are given in a ratio m : n, write them as mk and nk for an unknown scale factor k, then use the product relation to solve for k first, and the sum relation to get the required coefficient.

Application: For x² − px + 54 = 0 with roots in ratio 2 : 3:

1. Let the roots be 2k and 3k, since they are in the ratio 2 : 3.

2. By Vieta's formula, product of roots = constant term ÷ leading coefficient = 54 ÷ 1 = 54, so (2k)(3k) = 54, i.e. 6k2 = 54, giving k2 = 9, so k = ±3. By convention, when two quantities are stated to be in a ratio such as 2 : 3, both are taken as positive multiples of that ratio (a ratio does not, by itself, fix a sign) — so the accepted scale factor is k = 3.

3. By Vieta's formula, sum of roots = −(coefficient of x) ÷ leading coefficient = −(−p) ÷ 1 = p, so 2k + 3k = p, i.e. 5k = p.

4. Substituting k = 3: p = 5 × 3 = 15.

Cross-check: With k = 3 the roots are 2 × 3 = 6 and 3 × 3 = 9. Their product is 6 × 9 = 54, matching the constant term, and their sum is 6 + 9 = 15, matching p — both Vieta relations check out independently. (Taking k = −3 would give roots −6 and −9, also in ratio 2 : 3 with the same product 54, but sum −15; that value is not among the offered options. Exam questions asking for "the value of p" intend the single positive-scale-factor solution, and the option set here offers only 15, not −15 — corroborating the positive-ratio convention used above.)