Question

Let \(x + iy = (2 - i)^2 + \left(\frac{7-4i}{2-i}\right) - 8\), the values of \(x\) and \(y\) are:

• A. \(3, 7\)
• B. \(-3, 7\)
• C. \(3, -7\)
• D. \(-3, -7\)

Hint:

If your complex algebra yields unexpected decimals, look for sign typos in the question.
Changing a plus to a minus in denominators is a very common test error.
Solving with the corrected sign leads to the correct integer values: \(x = -3, y = -7\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To find the real values \(x\) and \(y\), we expand and simplify the complex expression on the right-hand side.
Once the expression is in the standard form \(a + ib\), we can equate the real and imaginary parts.

Step 3: Detailed Explanation:
Let us simplify each term of the expression step-by-step:
1. Simplify the first term \((2 - i)^2\):
\[ (2 - i)^2 = 2^2 - 2(2)(i) + i^2 \]
Since \(i^2 = -1\):
\[ (2 - i)^2 = 4 - 4i - 1 = 3 - 4i \]
2. Evaluate the rational fraction term:
\[ \frac{7 - 4i}{2 - i} \]
Wait, let us analyze the arithmetic. If we evaluate this term directly:
Multiply the numerator and the denominator by the conjugate of the denominator, \(2 + i\):
\[ \frac{7-4i}{2-i} \times \frac{2+i}{2+i} = \frac{14 + 7i - 8i - 4i^2}{2^2 - i^2} = \frac{14 - i + 4}{5} = \frac{18 - i}{5} = 3.6 - 0.2i \]
Substituting this back into the original expression:
\[ x + iy = (3 - 4i) + (3.6 - 0.2i) - 8 = -1.4 - 4.2i \]
This does not yield integer values for \(x\) and \(y\).
This indicates a typographical error in the original question's printed signs.
Let us analyze the common typo where the sign in the denominator of the fraction is positive: \(\frac{7 - 4i}{2 + i}\).
Evaluate this revised term:
Multiply the numerator and denominator by the conjugate \(2 - i\):
\[ \frac{7-4i}{2+i} \times \frac{2-i}{2-i} = \frac{14 - 7i - 8i + 4i^2}{2^2 - i^2} = \frac{14 - 15i - 4}{5} = \frac{10 - 15i}{5} = 2 - 3i \]
Now, substitute this clean integer term back into the equation:
\[ x + iy = (3 - 4i) + (2 - 3i) - 8 \]
Combine the real terms and imaginary terms:
\[ x + iy = (3 + 2 - 8) + i(-4 - 3) \]
\[ x + iy = -3 - 7i \]
Equating the real and imaginary parts yields:
\[ x = -3 \]
\[ y = -7 \]
This matches Option (D).

Step 4: Final Answer:
Therefore, the correct option is (D).