To solve $(2 + \sqrt{3})^2$ quickly and efficiently, we use the binomial square formula:

$(a + b)^2 = a^2 + 2ab + b^2$

Here:

• $a = 2$
• $b = \sqrt{3}$

Substitute the values:

$(2 + \sqrt{3})^2 = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2$

Step-by-step:

1. $2^2 = 4$
2. $2(2)(\sqrt{3}) = 4\sqrt{3}$
3. $(\sqrt{3})^2 = 3$

Add them up:

$4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$

Final Answer:

$(2 + \sqrt{3})^2 = 7 + 4\sqrt{3}$

Would you like further explanation or have additional questions?

Related Questions:

1. How do we expand a general binomial expression $(a + b)^n$?
2. What happens if the expression is $(2 - \sqrt{3})^2$?
3. How do we compute higher powers like $(2 + \sqrt{3})^3$?
4. Can you represent $\sqrt{3}$ approximately in the decimal form in the answer?
5. Why is the binomial expansion faster than directly multiplying?

Tip:

Memorizing the binomial square formula saves time for quick expansions in exams!