Construct the quadratic equation whose roots are √2 and 2√2. Express the equation with integer constant term and a rational middle coefficient if possible.

Introduction / Context:
Building a quadratic from its roots is a classic application of Vieta's relations. If r1 and r2 are the roots, then the monic quadratic is x^2 − (r1 + r2)x + r1 r2 = 0. The task uses irrational roots √2 and 2√2, but the product becomes an integer, simplifying the constant term.

Given Data / Assumptions:

• Roots: √2 and 2√2.
• Monic quadratic form: x^2 − (sum)x + product = 0.

Concept / Approach:
Compute the sum and product of the given roots, and substitute into the monic quadratic template. Double-check arithmetic with radicals to avoid mistakes.

Step-by-Step Solution:
Sum = √2 + 2√2 = 3√2Product = √2 * 2√2 = 2*(√2*√2) = 2*2 = 4Quadratic: x^2 − (3√2)x + 4 = 0

Verification / Alternative check:
Substitute x = √2: LHS = 2 − 3√2*√2 + 4 = 2 − 6 + 4 = 0. Substitute x = 2√2: LHS = 8 − 3√2*2√2 + 4 = 8 − 12 + 4 = 0. Both roots satisfy the equation.

Why Other Options Are Wrong:
Sign errors on the middle term or constant produce equations that do not vanish at both roots; testing either root quickly reveals nonzero residuals.

Common Pitfalls:
Miscomputing the product of radicals (√2 * 2√2 = 4, not 2√4). Keep the rule √2*√2 = 2 in mind.

Final Answer:
x^2 − 3√2 x + 4 = 0