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

Consider the quadratic equation \[ x^2 - 5x + k = 0. \] For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant is \[ D = b^2 - 4ac. \] Here, \(a = 1\), \(b = -5\), \(c = k\), so \[ D = 25 - 4k. \] Roots are \[ x = \frac{5 \pm \sqrt{25 - 4k}}{2}. \] For the roots to be integers: 1. The discriminant \(25 - 4k\) must be a non-negative perfect square. 2. The numerator \(5 \pm \sqrt{25 - 4k}\) must be even (so that division by 2 gives an integer). Step 1: Range of $k$ from non-negative discriminant. We need \[ 25 - 4k \ge 0 \Rightarrow 4k \le 25 \Rightarrow k \le 6.25. \] Since \(k\) is a non-negative integer, \[ k \in \{0,1,2,3,4,5,6\}. \]
Step 2: Check for perfect square discriminants. Compute \(D = 25 - 4k\) for each possible \(k\): \[ \begin{aligned} k = 0 &: D = 25 \quad (=5^2)
k = 1 &: D = 21 \quad (\text{not a perfect square})
k = 2 &: D = 17 \quad (\text{not a perfect square})
k = 3 &: D = 13 \quad (\text{not a perfect square})
k = 4 &: D = 9 \quad (=3^2)
k = 5 &: D = 5 \quad (\text{not a perfect square})
k = 6 &: D = 1 \quad (=1^2) \end{aligned} \] Thus, \(D\) is a perfect square for \[ k = 0,\ 4,\ 6. \]
Step 3: Verify roots are integers in these cases. - For \(k = 0\): \[ x^2 - 5x = 0 \Rightarrow x(x - 5) = 0 \Rightarrow x = 0, 5\quad (\text{integers}). \] - For \(k = 4\): \[ x^2 - 5x + 4 = 0 \Rightarrow x = \frac{5 \pm \sqrt{9}}{2} = \frac{5 \pm 3}{2} \Rightarrow x = 4, 1\quad (\text{integers}). \] - For \(k = 6\): \[ x^2 - 5x + 6 = 0 \Rightarrow x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} \Rightarrow x = 3, 2\quad (\text{integers}). \] All three values of \(k\) give integer roots. Therefore, the number of non-negative integer values of \(k\) for which the quadratic has only integer roots is \[ \boxed{3}. \]