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The problem asks to find a relationship between \(a\) and \(b\) such that the equations \(x^2 - ax + b = 0\) and \(x^2 - bx + a = 0\) have a common root.

### Step-by-Step Solution:

Let the common root be \( \alpha \).

#### Step 1: Substitute \( \alpha \) into both equations.

From the first equation:
\[
\alpha^2 - a\alpha + b = 0 \quad \text{(Equation 1)}
\]

From the second equation:
\[
\alpha^2 - b\alpha + a = 0 \quad \text{(Equation 2)}
\]

#### Step 2: Subtract Equation 2 from Equation 1 to eliminate \( \alpha^2 \):

\[
(\alpha^2 - a\alpha + b) - (\alpha^2 - b\alpha + a) = 0
\]

Simplifying, we get:
\[
-b\alpha + a\alpha + b - a = 0
\]

\[
(a - b)\alpha + (b - a) = 0
\]

\[
(a - b)(\alpha - 1) = 0
\]

#### Step 3: Analyze the result.

For the above equation to hold true, either:
1. \(a - b = 0\) which implies \(a = b\), or
2. \( \alpha - 1 = 0 \) which implies \( \alpha = 1 \).

If \( a = b \), both quadratic equations become identical, and hence they trivially have all roots in common. This is the trivial solution.

#### Step 4: Substitute \( \alpha = 1 \) into either equation.

Substituting \( \alpha = 1 \) into Equation 1:
\[
1^2 - a \cdot 1 + b = 0
\]

\[
1 - a + b = 0 \quad \Rightarrow \quad b = a - 1
\]

Thus, the non-trivial relationship between \(a\) and \(b\) is:
\[
b = a - 1
\]

### Conclusion:
The relationship between \(a\) and \(b\) such that the equations \(x^2 - ax + b = 0\) and \(x^2 - bx + a = 0\) have a common root is \( b = a - 1 \).

Would you like a detailed explanation of any step or have any other questions?

### Relative Questions:
1. What are the possible values of \(a\) and \(b\) if \(x^2 - ax + b = 0\) and \(x^2 - bx + a = 0\) share both roots?
2. How does the discriminant play a role in finding the roots of quadratic equations?
3. What happens if both roots of the given equations are common, not just one?
4. Can we generalize this result to other forms of quadratic equations?
5. What are the graphical interpretations of these two quadratic equations?

### Tip:
Always consider both trivial and non-trivial cases when dealing with conditions of common roots in polynomial equations.

Explore how to find a relationship between variables in quadratic equations x^2 - ax + b = 0 and x^2 - bx + a = 0. Learn the step-by-step solution and conditions under which these equations share a common root. Suitable for advanced high school students.

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