The number of non-negative integer values of k for which the quadratic equation x² – 5x + k = 0 has only integer roots, is

The correct answer is 3.

We are given the quadratic equation:

$x^2-5x+k=0$

Let the roots of this quadratic equation be represented by the Greek letters α and β. We are given that both roots are integers. Using the relationships between roots and coefficients (Vieta's formulas), we have the following:

Sum of roots:

$\alpha +\beta =5$

Product of roots:

$\alpha \beta =k$

We are also given that the constant is a non-negative integer, so:

$k\ge 0$

Since the product of the roots is non-negative, the roots must either both be non-negative or both be non-positive. However, since their sum is positive (5), both roots must be non-negative integers.

Let's list the possible pairs of non-negative integers that add up to 5, and find the corresponding value for the product:

Case 1: The roots are 0 and 5.

Then the product is:

$k=0\times 5=0$

Case 2: The roots are 1 and 4.

Then the product is:

$k=1\times 4=4$

Case 3: The roots are 2 and 3.

Then the product is:

$k=2\times 3=6$

Any other integer pairs will just be permutations of the above (for example, 3 and 2) and will yield the same values for the product. Thus, the possible values are 0, 4, and 6.

Since there are exactly 3 such values, the number of non-negative integer values for the constant is 3.