Find The quadratic Equation Whose Roots Are -2q And 5q.

Analyze the question: We need a quadratic equation whose roots (solutions) are x = -2q and x = 5q.

Concept: If the roots are α and β, the equation is:

x² - (α + β)x + αβ = 0

(sum of roots with a negative sign, then the product of roots)

Solution:

• Sum of roots: α + β = -2q + 5q = 3q
• Product of roots: αβ = (-2q)(5q) = -10q²

Substitute into the formula:

x² - (3q)x + (-10q²) = 0

x² - 3qx - 10q² = 0

Check using factors: (x + 2q)(x - 5q) = x² - 5qx + 2qx - 10q² = x² - 3qx - 10q² ✓

Comparing with the options:

• A: wrong, because it has a coefficient of 3 on x² and a +3qx term.
• B: wrong, since the signs of both the x term and the constant term are incorrect.
• C: has the right x term, but the constant term is +10q², when it should be -10q².
• D: x² - 3qx - 10q² matches our result.

Tip: Always remember "x² - (sum)x + (product)". A negative root times a positive root gives a negative product, so the constant term must be negative. That alone eliminates B and C immediately.