Given that `{:(A),(B):} = A^2 + B^2 + 2AB`, what is `A + B`, if `{:(A),(B):} = 9` ?

Answer

To solve the problem, we start with the given equation:

\[ \{(A),(B)\} = A^2 + B^2 + 2AB \]

We know that this expression can be rewritten using the identity for the square of a binomial:

\[ A^2 + B^2 + 2AB = (A + B)^2 \]

So we can rewrite the equation as:

\[ \{(A),(B)\} = (A + B)^2 \]

According to the problem, we have:

\[ \{(A),(B)\} = 9 \]

This means we can set up the equation:

\[ (A + B)^2 = 9 \]

Next, we will take the square root of both sides. Remember that when we take the square root, we must consider both the positive and negative roots:

\[ A + B = \sqrt{9} \quad \text{or} \quad A + B = -\sqrt{9} \]

Calculating the square root gives us:

\[ A + B = 3 \quad \text{or} \quad A + B = -3 \]

Thus, the final answer is:

\[ A + B = 3 \quad \text{or} \quad A + B = -3 \]