√(x + 45 – mn)4 = 5x + Q One root is x = –10, where m and n are the roots of…

Concept: For any real expression a, the principal square root of its fourth power is its square: √(a4) = a2. For a quadratic Y2 + bY + c = 0 with roots m and n, the product of roots mn = c (here c/a with a = 1). A "quantity comparison" then reduces to evaluating both quantities numerically and comparing.

1. Find mn from the quadratic. In Y2 – 11Y + 30 = 0, the product of roots mn = 30 (the roots are 5 and 6).

2. Simplify the left side using √(a4) = a2: √(x + 45 – mn)4 = (x + 45 – 30)2 = (x + 15)2.

3. The equation becomes (x + 15)2 = 5x + Q. Substitute the given root x = –10: (–10 + 15)2 = 5(–10) + Q.

4. Evaluate: (5)2 = –50 + Q, i.e. 25 = –50 + Q, so Q = 75.

5. Compute the quantities. Quantity I = Q/3 = 75/3 = 25. Quantity II = Q/22 = 75/4 = 18.75.

6. Compare: 25 > 18.75, therefore Quantity I > Quantity II.

Cross-check: Q is a single fixed value (75) that is positive, so dividing it by the smaller denominator (3) must give a larger result than dividing by the larger denominator (4). 75/3 = 25 and 75/4 = 18.75 confirm Quantity I exceeds Quantity II.