The roots of a quadratic equation in x, are -m and 2n. Fine equation.

Analyze the question: We are given the roots of a quadratic equation, -m and 2n, and must form the equation.

Concept: A quadratic equation with roots α and β is x² - (α + β)x + αβ = 0

Solution:

Sum of roots: α + β = -m + 2n = 2n - m

Product of roots: αβ = (-m)(2n) = -2mn

Substitute: x² - (2n - m)x + (-2mn) = 0 x² - x(2n - m) - 2mn = 0

Check against the options:

• Option A: x² - x(2n - m) - 2mn = 0, which matches exactly.
• Option B has + x(2n - m), so the sign of the middle term is wrong.
• Option C and D have a +2mn product term, which is wrong because the product is negative.
• Option E: x² - x(m - 2n) - 2mn = 0 has the middle term as -(m - 2n) = 2n... wait, that equals +(2n - m)x... Let us check: -x(m - 2n) = x(2n - m), which gives the wrong sign for the sum. So E is wrong.

Verify with numbers: Let m = 1 and n = 1. The roots are -1 and 2. Option A gives x² - x(2 - 1) - 2 = x² - x - 2 = (x - 2)(x + 1) = 0, so the roots are 2 and -1. ✔

Tip: Always use x² - (sum)x + (product) = 0. The most common mistake is forgetting the minus sign in front of the sum. When roots are given with letters, quickly test with simple numbers (like m = 1, n = 1) to confirm.