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Chapter 2: Problem 53

Perform the indicated operation(s) and write the result in standard form. $$(2+i)^{2}-(3-i)^{2}$$

Short Answer

Expert verified
The result of the operation \((2+i)^{2}-(3-i)^{2}\) in standard form is \( -5 +10i \).

Step by step solution

01

Squaring the Binomials

Start by squaring \((2+i)^{2}\) and \((3-i)^{2}\) separately. Remember to use the FOIL method. When squaring \((2+i)^{2}\), firstly multiply the First terms, then the Outside terms, then the Inside terms, and finally, the Last terms. This gives \((2+i)^{2} = 2^{2} + 2(2)(i) + (i)^{2}\). Similarly, squaring the other binomial \((3-i)^{2}\) gives \((3-i)^{2} = 3^{2} - 2(3)(i) + (-i)^{2}\). Simplify remembering that \(i^2 = -1\).
02

Simplifications and Subtraction

You obtain \((2+i)^{2} = 4 + 4i - 1\) and \((3-i)^{2} = 9 - 6i - 1\) when simplified. Now subtract: \( (4 + 4i - 1) - (9 - 6i - 1)\). Remember to distribute the negative sign to all terms inside the parenthesis.
03

Final Simplification

Performing the subtraction operation gives: \( 4 -9 + 4i + 6i - 1 +1 = -5 +10i \). This is the final result in standard form for complex numbers, which is \( a + bi\).

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