Write $(5-5 i)^2$ in the form $a+b i$, where $a$ and $b$ are real numbers.

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daisytheanimallover1 daisytheanimallover1

05-11-2026
Mathematics

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Write $(5-5 i)^2$ in the form $a+b i$, where $a$ and $b$ are real numbers.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T20:42:59-05:00

Apply the binomial expansion identity $(x-y)^2 = x^2 - 2xy + y^2$ to $(5-5i)^2$.
Calculate the terms: $5^2 = 25$, $-2(5)(5i) = -50i$, and $(5i)^2 = 25i^2 = -25$.
Combine the terms to find $-50i$.
Express the result in the standard form $a+bi$: $\boxed{0-50i}$.

Explanation

Problem Analysis

The problem asks to expand the expression $(5-5i)^2$ and express it in the standard form of a complex number, $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit defined by $i^2 = -1$. This is an exercise in applying algebraic binomial expansion to complex numbers.

Expanding the Expression

To expand the squared binomial $(5-5i)^2$, we use the algebraic identity $(x-y)^2 = x^2 - 2xy + y^2$. Here, let $x = 5$ and $y = 5i$. Substituting these values into the identity, we get:
\[(5-5i)^2 = 5^2 - 2(5)(5i) + (5i)^2\]
Now, we calculate each term individually:

$5^2 = 25$
$2(5)(5i) = 50i$
$(5i)^2 = 5^2 \times i^2 = 25 \times (-1) = -25$

Combining these, we have:
\[(5-5i)^2 = 25 - 50i - 25\]
Subtracting the constant terms, we find:
\[(5-5i)^2 = -50i\]
This can be written in the form $a+bi$ as $0-50i$.

Conclusion

The final result of the expansion is $-50i$. In the standard complex form $a+bi$, we have $a = 0$ and $b = -50$. Thus, $(5-5i)^2 = \boxed{0-50i}$.

Examples

Complex number multiplication and exponentiation are essential in fields like electrical engineering and physics, particularly when analyzing alternating current (AC) circuits or wave phenomena. For example, when calculating the impedance of a circuit, engineers use complex numbers to represent both resistance and reactance, allowing them to simplify complex oscillating systems into manageable algebraic equations. Mastering this allows you to solve problems involving phase shifts and signal propagation with ease.