Textbook Question
Find each product and write the result in standard form. (7 - 5i)(- 2 - 3i)
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1. Equations & Inequalities
The Imaginary Unit
2:25 minutesProblem 20
Textbook Question
In Exercises 9–20, find each product and write the result in standard form. (5 - 2i)2
Verified step by step guidance1
Recognize that the expression (5 - 2i)^2 is a binomial squared. Use the formula for the square of a binomial: (a - b)^2 = a^2 - 2ab + b^2.
Identify the values of a and b in the binomial. Here, a = 5 and b = 2i.
Substitute the values of a and b into the formula: (5 - 2i)^2 = 5^2 - 2(5)(2i) + (2i)^2.
Simplify each term: 5^2 = 25, -2(5)(2i) = -20i, and (2i)^2 = 4i^2. Recall that i^2 = -1, so 4i^2 = 4(-1) = -4.
Combine the simplified terms: 25 - 20i - 4. Write the result in standard form as a + bi, where a is the real part and b is the imaginary part.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.
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Multiplication of Complex Numbers
To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) and combine like terms. When multiplying, remember that i^2 equals -1, which is crucial for simplifying the result. This process allows you to find the product of two complex numbers effectively.
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Standard Form of Complex Numbers
The standard form of a complex number is expressed as a + bi, where a and b are real numbers. In this form, a represents the real part, and bi represents the imaginary part. Writing complex numbers in standard form is important for clarity and consistency in mathematical communication, especially when performing further operations.
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