For which value of k does the equation x^2 - 4x + k = 0 have roots that differ by 2?

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Q: For which value of k does the equation x^2 - 4x + k = 0 have roots that differ by 2?

Solution: Let the roots be r and r+2. Then, r + (r+2) = 4 and r(r+2) = k leads to k = 4.

Steps: 9

Step 1: Understand that we need to find a value of k for the equation x^2 - 4x + k = 0.
Step 2: Recognize that the roots of the equation are the values of x that make the equation equal to zero.
Step 3: Let the two roots be r (the first root) and r + 2 (the second root, which is 2 more than the first).
Step 4: Use the property of roots: the sum of the roots (r + (r + 2)) should equal the coefficient of x (which is -(-4) = 4).
Step 5: Set up the equation: r + (r + 2) = 4. This simplifies to 2r + 2 = 4.
Step 6: Solve for r: Subtract 2 from both sides to get 2r = 2, then divide by 2 to find r = 1.
Step 7: Now find the second root: r + 2 = 1 + 2 = 3.
Step 8: Use the product of the roots to find k: The product of the roots (r * (r + 2)) should equal k.
Step 9: Calculate the product: 1 * 3 = 3, so k = 3.