Question

The number of non-negative integer values of $k$ for which the quadratic equation $x^2 - 5x + k = 0$ has only integer roots, is:

Hint:

For quadratics with integer coefficients, to ensure integer roots: \begin{itemize} \item The discriminant must be a non-negative perfect square. \item Check that the resulting expression for the roots truly gives integers (often a parity check on the numerator). \end{itemize}

Answer 1

Correct Answer: 3

Step 1: If the roots are integers \(r\) and \(s\), then \(r+s=5\) and \(rs=k\). Write \(s = 5-r\), so \(k = r(5-r)\).
Step 2: We need \(k \ge 0\), i.e. \(r(5-r) \ge 0\), which holds for \(0 \le r \le 5\). Checking each integer in this range: \(r=0\Rightarrow k=0\); \(r=1\Rightarrow k=4\); \(r=2\Rightarrow k=6\); \(r=3\Rightarrow k=6\); \(r=4\Rightarrow k=4\); \(r=5\Rightarrow k=0\).
Step 3: The distinct values of \(k\) obtained are \(\{0, 4, 6\}\).
\[ \boxed{3} \]