Question

If the roots of the equation $x^2 - kx + 16 = 0$ are integers, what is the sum of the roots? 
 

• A. 4
• B. 6
• C. 8
• D. 10

Hint:

For quadratics with integer roots, list factor pairs of the constant term and check sums against options.

Answer 1

The Correct Option is C

- Step 1: For $x^2 - kx + 16 = 0$, sum of roots = $k$, product = 16. Roots are integers $a, b$ such that $a \cdot b = 16$.
- Step 2: Possible pairs: $(1, 16)$, $(2, 8)$, $(4, 4)$, $(-1, -16)$, $(-2, -8)$, $(-4, -4)$.
- Step 3: Sum of roots: $1 + 16 = 17$, $2 + 8 = 10$, $4 + 4 = 8$, $-1 - 16 = -17$, $-2 - 8 = -10$, $-4 - 4 = -8$.
- Step 4: Check options: Option (c) is 8, corresponding to roots $4, 4$.
- Step 5: Verify: If roots are 4, 4, equation is $(x - 4)^2 = x^2 - 8x + 16$, so $k = 8$.
- Step 6: Other pairs don't match options, so option (c) is correct.