Question 22

25

The expression (3 − 4i)(3 + 4i) simplifies to 25 by using the difference of squares formula, where a^2 - b^2 results in real numbers when complex numbers are involved.

A) −25

This option is incorrect because the calculation of (3 − 4i)(3 + 4i) does not yield a negative result. Applying the difference of squares gives a positive result, specifically 9 + 16, which equals 25.

B) −7

This option is also incorrect as it does not reflect the outcome of the multiplication. The result of (3 − 4i)(3 + 4i) is a positive number, not a negative one, and certainly not as low as -7.

C) 25

This option is not chosen as the correct answer in the context provided. However, it is the same value as the correct answer, showing that while it is a plausible option, it is not the one designated as correct in this question.

D) 25

This is the correct answer as the calculation of (3 − 4i)(3 + 4i) results in 25. By applying the difference of squares, we find that 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25.

E) 7

This option is incorrect as it underestimates the result of the multiplication. The operation yields a value significantly higher than 7, as demonstrated through the difference of squares approach.

Conclusion

The correct answer is 25, which is confirmed through the proper application of the difference of squares formula. All other options are incorrect as they do not accurately reflect the mathematical outcome of the expression, which results in a positive integer rather than a negative or smaller positive value.