Construct a quadratic from sum and product of roots: Form the quadratic equation whose roots have sum 6 and product −16.

Introduction / Context:Given the sum (S) and product (P) of roots, the monic quadratic is x^2 − Sx + P = 0. This is a direct application of Vieta’s relations and avoids computing the individual roots.Given Data / Assumptions:

• Sum S = 6.
• Product P = −16.

Concept / Approach:Write x^2 − Sx + P = 0. Substitute S and P as provided, ensuring the signs are placed correctly (note that P is negative here).

Step-by-Step Solution:

Verification / Alternative check:If roots are r1 and r2, then r1 + r2 = 6 and r1*r2 = −16. Expanding (x − r1)(x − r2) gives x^2 − (r1 + r2)x + r1r2 = x^2 − 6x − 16, as required.

Why Other Options Are Wrong:

• x^2 + 6x − 16 = 0: Has sum −6.
• x^2 − √3x − 16 = 0: Introduces an irrelevant irrational coefficient.
• None of these: Incorrect because x^2 − 6x − 16 = 0 matches exactly.

Common Pitfalls:Mixing the sign of P or writing x^2 − Sx − P by mistake. Always align with x^2 − Sx + P.

Final Answer: