Q7
The number of non-negative integer values of \(k\) for which the quadratic equation \(x^2-5 x+k=0\) has only integer roots, is
Answer: 3
Key idea: If the roots are integers \(p, q\), then \(p+q=5\) and \(pq=k\ge0\).
- \(pq\ge0\) means both roots have the same sign (or one is zero). Two negative numbers cannot add up to 5, so both roots are non-negative.
- Non-negative pairs with sum 5: \((0,5),(1,4),(2,3)\).
- These give \(k=0,4,6\).
Tip: Equivalently, the discriminant \(25-4k\) must be a perfect square; \(k=0,4,6\) give 25, 9, 1.
Hence, the answer is 3.