Two quadratics share a common root: x^2 + 2x − 3 = 0 and x^2 + 3x − k = 0. Find the non-zero value of k.

Introduction / Context:If two quadratics share a root r, then r satisfies both equations. Solve the first to get its roots, test them in the second, and find k that makes the second equation zero for that r. The question specifies the non-zero k if there is a choice.

Given Data / Assumptions:

• Eqn1: x^2 + 2x − 3 = 0
• Eqn2: x^2 + 3x − k = 0
• Common real root exists; choose non-zero k.

Concept / Approach:Factor Eqn1 to get candidate roots. Substitute each into Eqn2 to solve for k. If multiple k values occur, follow the “non-zero” instruction.

Step-by-Step Solution:Eqn1 factors: (x + 3)(x − 1) = 0 ⇒ roots r ∈ {−3, 1}Substitute r = 1 into Eqn2: 1 + 3 − k = 0 ⇒ k = 4Substitute r = −3 into Eqn2: 9 − 9 − k = 0 ⇒ k = 0 (disallowed as per prompt)Hence the non-zero k is 4

Verification / Alternative check:With k = 4, Eqn2 becomes x^2 + 3x − 4 = 0 = (x + 4)(x − 1); root x = 1 matches Eqn1.

Why Other Options Are Wrong:1, 2, 3 do not produce a shared root with Eqn1; only 4 works and is non-zero.

Common Pitfalls:Missing the “non-zero” qualifier and selecting k = 0. Always read such constraints carefully.

Final Answer:4