The number of non-negative integer values of $k$ for which the quadratic equation $x^2 - 5x + k = 0$ has only integer roots, is:
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⚡ AZUCATION SHORTCUT:
For $x^2 - Sx + P = 0$, if roots are integers, their sum is $5$ and product is $k$.
Possible pairs $(\alpha, \beta)$ summing to 5: $(0,5), (1,4), (2,3)$.
Since $k = \alpha\beta$, find unique values of $k$.
Detailed Step-by-Step Solution
Step 1: Understand the Properties of Roots
Let the roots of the equation $x^2 - 5x + k = 0$ be $\alpha$ and $\beta$.
From the relation between roots and coefficients:
Sum of roots: $\alpha + \beta = -(-5)/1 = 5$
Product of roots: $\alpha \cdot \beta = k/1 = k$
Step 2: List Integer Pairs
We are looking for pairs of integers $(\alpha, \beta)$ such that their sum is 5 and their product $k$ is a non-negative integer ($k \ge 0$).
Possible integer pairs $(\alpha, \beta)$ where $\alpha + \beta = 5$:
* $0 + 5 = 5 \implies k = 0 \times 5 = 0$
* $1 + 4 = 5 \implies k = 1 \times 4 = 4$
* $2 + 3 = 5 \implies k = 2 \times 3 = 6$
* $-1 + 6 = 5 \implies k = -6$ (Invalid, $k$ must be non-negative)
* Other negative combinations will also result in a negative product $k$.
Step 3: Count the Values of $k$
The possible non-negative integer values for $k$ are $\{0, 4, 6\}$.
Counting these values, we get exactly 3 values.
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