Which complex number is equivalent to the given expression?
$(-45-22 i)+2(5-3 i)(5+3 i)$
The problem requires simplifying the complex expression \((-45 - 22i) + 2(5 - 3i)(5 + 3i)\). Recall that the standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit satisfying \(i^2 = -1\). We will simplify the expression step by step using basic algebraic operations and properties of complex numbers.
First, consider the product of the complex numbers \((5 - 3i)(5 + 3i)\). This is a product of a complex number and its conjugate, which follows the pattern \((x - yi)(x + yi) = x^2 + y^2\). Applying this formula, we get:
\[(5 - 3i)(5 + 3i) = 5^2 + 3^2 = 25 + 9 = 34\]
Now, substitute this back into the original expression, which becomes:
\[(-45 - 22i) + 2(34)\]
Multiplying 34 by 2 gives 68, so the expression simplifies further to:
\[(-45 - 22i) + 68\]
Finally, combine the real parts of the complex number. The real part is \(-45 + 68\), and the imaginary part remains \(-22i\).
\[-45 + 68 = 23\]
Thus, the simplified form of the entire expression is:
\[23 - 22i\]
Comparing this result with the provided options, we find it corresponds to the second option.
Complex numbers are essential in electrical engineering for analyzing alternating current (AC) circuits. When you model the impedance of resistors, capacitors, and inductors as complex numbers, you can easily combine them to calculate the total behavior of the circuit using algebra. For instance, calculating the voltage across a component in an AC circuit often involves operations like the one solved here, allowing engineers to design everything from smartphone signal processing to electrical power grids.