Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha\beta = \frac{c}{a}$.
• Difference of Roots: The square of the difference of roots can be expressed as $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$.
• Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$.

Step-by-Step Solution

1. Analyze the first quadratic equation Let the first quadratic equation be $x^2 + ax + b = 0$. Let its roots be $\alpha$ and $\beta$.

• Applying Vieta's formulas:
  • The sum of the roots is $\alpha + \beta = -a$.
  • The product of the roots is $\alpha\beta = b$.
• Calculating the square of the difference of roots for the first equation: Using the identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$: Substituting the values from Vieta's formulas: $(\alpha - \beta)^2 = (-a)^2 - 4(b)$ $(\alpha - \beta)^2 = a^2 - 4b$ This expression represents the square of the difference between the roots of the first equation in terms of $a$ and $b$.

2. Analyze the second quadratic equation Let the second quadratic equation be $x^2 + bx + a = 0$. Let its roots be $\gamma$ and $\delta$.

• Applying Vieta's formulas:
  • The sum of the roots is $\gamma + \delta = -b$.
  • The product of the roots is $\gamma\delta = a$.
• Calculating the square of the difference of roots for the second equation: Using the identity $(\gamma - \delta)^2 = (\gamma + \delta)^2 - 4\gamma\delta$: Substituting the values from Vieta's formulas: $(\gamma - \delta)^2 = (-b)^2 - 4(a)$ $(\gamma - \delta)^2 = b^2 - 4a$ This expression represents the square of the difference between the roots of the second equation in terms of $a$ and $b$.

3. Use the given condition to form an equation The problem states that the difference between the corresponding roots of the two equations is the same. This means: $|\alpha - \beta| = |\gamma - \delta|$ Squaring both sides of this equality: $(\alpha - \beta)^2 = (\gamma - \delta)^2$ Substituting the expressions derived in Step 1 and Step 2: $a^2 - 4b = b^2 - 4a$

4. Solve for the relationship between 'a' and 'b' Rearrange the equation $a^2 - 4b = b^2 - 4a$ to find the required relationship. Move all terms to one side to set the equation to zero: $a^2 - b^2 - 4b + 4a = 0$ Rearrange the terms: $(a^2 - b^2) + (4a - 4b) = 0$ Factor each group:

• The first group is a difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
• The second group has a common factor of 4: $4a - 4b = 4(a-b)$. Substitute these factored forms back into the equation: $(a-b)(a+b) + 4(a-b) = 0$ Factor out the common factor $(a-b)$: $(a-b)(a+b+4) = 0$ For this product to be zero, at least one of the factors must be zero. So, either $(a-b) = 0$ or $(a+b+4) = 0$. The problem statement explicitly mentions that $a \ne b$. This implies that $a-b \ne 0$. Therefore, the other factor must be zero: $a+b+4 = 0$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when applying Vieta's formulas and simplifying expressions. Forgetting the negative sign in the sum of the roots is a common mistake.
• Factoring: Correctly factoring the equation is crucial. Make sure to identify and factor out the common terms.
• Using all information: Remember to use the condition $a \ne b$ to eliminate the $a-b = 0$ solution.

Summary

This problem demonstrates how to use Vieta's formulas and the difference of roots identity to relate the coefficients of two quadratic equations, given a condition about their roots. By setting up an equation based on the given condition and then carefully factoring, we arrive at the relationship $a+b+4 = 0$.

Final Answer

The final answer is $\boxed{a+b+4=0}$, which corresponds to option (A).