Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 2 + √-4)2
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1. Equations & Inequalities
The Imaginary Unit
1:28 minutesProblem 60a
Textbook Question
Find each product. Write answers in standard form. (2+i)²
Verified step by step guidance1
Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).
Identify \(a = 2\) and \(b = i\) in the expression \((2 + i)^2\).
Apply the formula: \((2 + i)^2 = 2^2 + 2 \cdot 2 \cdot i + i^2\).
Calculate each term separately: \$2^2 = 4\(, \(2 \cdot 2 \cdot i = 4i\), and recall that \)i^2 = -1$.
Combine the terms to write the expression in standard form: \$4 + 4i + (-1)\(, then simplify to \)3 + 4i$.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers
Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding how to work with complex numbers is essential for operations like addition, multiplication, and exponentiation.
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Binomial Expansion
Binomial expansion involves expanding expressions raised to a power, such as (a + b)² = a² + 2ab + b². This formula helps simplify powers of binomials, including those with complex terms, by systematically multiplying and combining like terms.
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The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Writing answers in standard form means expressing the result clearly with real and imaginary parts separated, which is important for clarity and further calculations.
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