Evaluate (2√3 - 4) (2√3 + 4)

Analysis

The expression $(2\sqrt{3} - 4)(2\sqrt{3} + 4)$ is a classic example of the Difference of Two Squares algebraic identity.

Concept

The difference of two squares formula states that:

In this problem:

• $a = 2\sqrt{3}$
• $b = 4$

Solution

Applying the formula $a^2 - b^2$:

1. Square the first term ($a^2$): $(2\sqrt{3})^2 = 2^2 \times (\sqrt{3})^2$ $= 4 \times 3$ $= 12$

2. Square the second term ($b^2$): $(4)^2 = 16$

3. Subtract the second from the first: $12 - 16 = -4$

Alternatively, you can use the FOIL method (First, Outer, Inner, Last):

• First: $2\sqrt{3} \times 2\sqrt{3} = 12$
• Outer: $2\sqrt{3} \times 4 = 8\sqrt{3}$
• Inner: $-4 \times 2\sqrt{3} = -8\sqrt{3}$
• Last: $-4 \times 4 = -16$

Combine them: $12 + 8\sqrt{3} - 8\sqrt{3} - 16$ The middle terms ($8\sqrt{3} - 8\sqrt{3}$) cancel out to zero, leaving: $12 - 16 = -4$

Conclusion

The result of the evaluation is -4, which corresponds to option A.

Exam Tip

Whenever you see two brackets with the same terms but different signs (e.g., $(x-y)(x+y)$), don't waste time multiplying every single term. Simply square the first term, square the second term, and put a minus sign between them. It saves time and reduces the risk of calculation errors!