1. **State the problem:** We are given the equation $$(3 - X)(3 + X) = a + x^2$$ and need to find the value of $a$. 2. **Recall the formula:** The expression on the left is a product of conjugates, which follows the difference of squares formula: $$ (A - B)(A + B) = A^2 - B^2 $$ 3. **Apply the formula:** Here, $A = 3$ and $B = X$, so $$ (3 - X)(3 + X) = 3^2 - X^2 = 9 - X^2 $$ 4. **Rewrite the equation:** Substitute back into the original equation: $$ 9 - X^2 = a + x^2 $$ 5. **Compare terms:** Group like terms: $$ 9 - X^2 = a + x^2 $$ 6. **Isolate $a$:** Add $X^2$ to both sides: $$ 9 = a + x^2 + X^2 $$ Since $x^2$ and $X^2$ represent the same term (case insensitive), this simplifies to: $$ 9 = a + 2x^2 $$ 7. **Check the original problem:** The problem likely intends $x$ and $X$ to be the same variable, so the original equation is: $$ (3 - x)(3 + x) = a + x^2 $$ From step 3, the left side is: $$ 9 - x^2 $$ Set equal to the right side: $$ 9 - x^2 = a + x^2 $$ 8. **Solve for $a$:** Add $x^2$ to both sides: $$ 9 = a + 2x^2 $$ Since this must hold for all $x$, the coefficient of $x^2$ on the right must be zero, so: $$ 2x^2 = 0 \implies x^2 = 0 $$ This is only true if $x=0$, but the equation must hold for all $x$, so the only way is if the $x^2$ terms cancel out, meaning: $$ a + x^2 = 9 - x^2 \implies a = 9 - 2x^2 $$ But since $a$ is a constant, the $x^2$ term must be zero, so: $$ a = 9 $$ **Final answer:** $$ a = 9 $$