Let r ∈ Z be a nonzero solution of
x2 + ax + b = 0
where a, b ∈ Z. Prove that r | b

Let's prove that if \( r \) is a nonzero integer solution to the quadratic equation \( x^2 + ax + b = 0 \), where \( a \) and \( b \) are integers, then \( r \) must divide \( b \).

### Proof:

Given the quadratic equation:
\[
x^2 + ax + b = 0
\]
Let's assume that \( r \) is a nonzero solution to this equation. Therefore, we can substitute \( r \) into the equation:
\[
r^2 + ar + b = 0
\]
Rearranging the equation, we get:
\[
r^2 + ar = -b
\]
Factor the left-hand side:
\[
r(r + a) = -b
\]
Since \( r \) is a nonzero integer, it follows that \( r \) divides the product \( r(r + a) \). Therefore, \( r \) must also divide \( -b \), because \( r \) is an integer divisor on the left-hand side.

Since \( r \) divides \( -b \), it also divides \( b \) (because divisibility by an integer remains unaffected by a sign change).

Thus, we have proved that \( r \) divides \( b \).

### Conclusion:
If \( r \) is a nonzero integer solution to the equation \( x^2 + ax + b = 0 \), then \( r \) must divide \( b \).

Would you like any more details or have any questions?

Here are some related questions you might find interesting:
1. How do we solve the quadratic equation \( x^2 + ax + b = 0 \) using the quadratic formula?
2. Can we extend the divisibility property to complex or irrational roots?
3. What happens when the discriminant of the quadratic equation is negative?
4. How does Vieta's formulas relate to the divisibility properties of the roots?
5. What are the conditions for both roots of the quadratic equation to be integers?

**Tip:** When dealing with quadratic equations, always consider the relationships between the coefficients and the roots as described by Vieta's formulas.

Let r ∈ Z be a nonzero solution of x^2 + ax + b = 0 where a, b ∈ Z. Prove that r | b

This tutorial guides you through proving that a nonzero integer root of the quadratic equation x^2 + ax + b = 0 must divide the constant term b. It covers algebraic concepts, including quadratic equations, Vieta's formulas, and divisibility rules. Suitable for students in Grades 9-12.

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