Correct Answer:
A. 3
The correct answer is A: 3. For a quadratic equation in the standard form ax² + bx + c = 0, let the roots be α and β. According to Vieta's formulas, the sum of the roots is α + β = -b/a, and the product of the roots is αβ = c/a.
In the given equation x² - 4x + k = 0, we have a = 1, b = -4, and c = k. Therefore:
- The sum of the roots: α + β = -(-4)/1 = 4.
- We are given that the roots differ by 2, which means |α - β| = 2. Squaring both sides gives (α - β)² = 4.
We use the algebraic identity: (α - β)² = (α + β)² - 4αβ.
Substituting the known values into the identity:
- 4 = (4)² - 4(k)
- 4 = 16 - 4k
- Rearranging the terms to solve for k:
- 4k = 16 - 4
- 4k = 12
- k = 12 / 4
- k = 3
Thus, the value of k is 3.
- B: -3 is incorrect. This result would occur if there was an arithmetic error, such as mismanaging the sign in the equation 4 = 16 - 4k, leading to 4k = -12 instead of 4k = 12.
- C: -1/3 and D: 1/3 are incorrect. These values are significantly different and would typically arise from substantial algebraic errors in applying the formulas, such as incorrect division, miscalculation of squares, or confusing the formulas for sum/product with other properties of roots. The systematic application of Vieta's formulas and the identity for the difference of roots leads directly to k = 3.