Verified Answer :

  • Use the FOIL method to expand the expression \((2-7i)(-1+4i)\).
  • Simplify the terms, replacing \(i^2\) with \(-1\).
  • Combine the resulting real and imaginary parts to obtain \(26+15i\).
  • Match the result to option (a), giving the final answer of \(\boxed{26+15i}\).

Explanation

  1. Expanding the Product of Complex Numbers

To calculate the product of two complex numbers \((2 - 7i)\) and \((-1 + 4i)\), we use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). This method ensures that each term in the first complex number is multiplied by each term in the second complex number.

The expression is:
\[ (2 - 7i)(-1 + 4i) \]

Applying the FOIL method:

  • First: Multiply the real parts: \(2 \times (-1) = -2\)
  • Outer: Multiply the outer terms: \(2 \times 4i = 8i\)
  • Inner: Multiply the inner terms: \((-7i) \times (-1) = 7i\)
  • Last: Multiply the imaginary parts: \((-7i) \times 4i = -28i^2\)
  1. Simplifying and Grouping Terms

Now, we combine the results from the FOIL expansion:
\[ -2 + 8i + 7i - 28i^2 \]

Next, we simplify the expression using the fundamental property of the imaginary unit, which states that \(i^2 = -1\). Substituting \(-1\) for \(i^2\):
\[ -28i^2 = -28 \times (-1) = 28 \]

Now the expression becomes:
\[ -2 + 8i + 7i + 28 \]

Grouping the real and imaginary parts:

  • Real part: \(-2 + 28 = 26\)
  • Imaginary part: \(8i + 7i = 15i\)

Combining these gives the final result:
\[ 26 + 15i \]

  1. Final Result and Conclusion

Comparing our result \(26 + 15i\) with the given options:

A\(26+15i\)
B\(26+i\)
C\(-30+15i\)
D\(-2-28i\)

The calculated result corresponds to option (a). Therefore, the final answer is \(\boxed{26+15i}\).

Examples

Complex numbers are essential in electrical engineering for analyzing alternating current (AC) circuits. When representing impedance or voltage as complex numbers, multiplying them allows engineers to determine the total power or the phase shift across various components like resistors, capacitors, and inductors. For instance, calculating the product of complex representations of current and voltage helps in finding the complex power in a circuit, similar to the multiplication performed here.

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