To solve for $a$ in the equation $(ax + 3y)^2 = a^2 x^2 - 6xy + 9y^2$, let's expand the left side and then compare it to the right side.

Step 1: Expand the left-hand side.

We have: $(ax + 3y)^2 = (ax + 3y)(ax + 3y)$

Expanding this: $(ax + 3y)(ax + 3y) = ax \cdot ax + ax \cdot 3y + 3y \cdot ax + 3y \cdot 3y$

This simplifies to: $a^2x^2 + 3axy + 3axy + 9y^2$

Combine like terms: $a^2x^2 + 6axy + 9y^2$

Step 2: Compare both sides of the equation.

We have now: $a^2x^2 + 6axy + 9y^2 = a^2x^2 - 6xy + 9y^2$

Step 3: Equate the coefficients of like terms.

• The coefficients of $x^2$ are the same on both sides: $a^2x^2$.
• For the $xy$ term, we have $6a = -6$.
• The coefficients of $y^2$ are the same on both sides: $9y^2$.

Step 4: Solve for $a$.

From $6a = -6$, we can solve for $a$: $a = \frac{-6}{6} = -1$

Final Answer:

$a = -1$

Would you like more details, or do you have any other questions?

Here are 5 related questions:

1. What happens if $a$ is a different value?
2. How would the solution change if the signs of terms were different?
3. Can you solve the equation using substitution or other methods?
4. How does this method relate to factoring?
5. What if the equation involved higher powers of $x$ or $y$?

Tip: Always compare the coefficients of corresponding terms when solving polynomial identities.