In algebra, the quadratic equation x^2 + k1·x + k2 = 0 is known to have (x − 2) and (x + 3) as its linear factors. Using this information, determine the exact values of the coefficients k1 and k2.

Introduction / Context:
This question checks your understanding of how the factorised form of a quadratic relates to its standard form x^2 + k1 x + k2. Once the roots or factors are known, you can simply expand their product to recover the coefficients. This technique is widely used when constructing polynomials from given roots and also when reverse engineering equations from factor information.

Given Data / Assumptions:
- The quadratic equation can be written as x^2 + k1 x + k2 = 0.
- Linear factors of this quadratic are given as (x − 2) and (x + 3).
- We need to find the constants k1 and k2 in terms of these factors.
- Standard algebraic expansion rules apply.

Concept / Approach:
If a quadratic has factors (x − α) and (x − β), then its expanded form is x^2 − (α + β)x + αβ. In this problem the factors are (x − 2) and (x + 3). Instead of trying to match this pattern immediately, the most straightforward way is to multiply the two binomials directly and then compare the resulting coefficients with x^2 + k1 x + k2.

Step-by-Step Solution:
1) Start with the factorised form: (x − 2)(x + 3). 2) Expand using the distributive property: x(x + 3) − 2(x + 3). 3) This gives x^2 + 3x − 2x − 6. 4) Combine like terms in the middle: x^2 + (3x − 2x) − 6 = x^2 + x − 6. 5) Compare with the standard form x^2 + k1 x + k2. 6) Clearly, k1 = 1 and k2 = −6.

Verification / Alternative check:
We can verify by substituting x = 2 and x = −3 into x^2 + x − 6. For x = 2, we get 4 + 2 − 6 = 0. For x = −3, we get 9 − 3 − 6 = 0. Both roots satisfy the quadratic, which confirms that the factorisation and coefficients are correct. Since the coefficient of x^2 is 1 and the two roots are fixed, there is no freedom to choose different values for k1 and k2.

Why Other Options Are Wrong:
Option B and Option C swap signs or swap places of coefficients incorrectly. Option D (k1 = 1, k2 = 6) would give the quadratic x^2 + x + 6, which does not have roots 2 and −3. Option E mixes both signs incorrectly. Only Option A matches exactly the coefficients obtained from the correct expansion of (x − 2)(x + 3).

Common Pitfalls:
A common mistake is incorrect expansion, such as x^2 + 5x − 6 if someone simply adds 2 and 3 without accounting for the minus sign. Another frequent issue is sign confusion when comparing with x^2 + k1 x + k2, especially when one or both roots are negative. Writing out every intermediate step clearly and checking by substitution helps avoid these simple but costly errors.

Final Answer:
Expanding the given factors and comparing coefficients shows that k1 = 1 and k2 = −6, so the correct choice is k1 = 1, k2 = -6.