Textbook Question
Find each product. Write answers in standard form. (2+i)(3-2i)
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1. Equations & Inequalities
Complex Numbers
1:52 minutesProblem 68a
Textbook Question
Find each product. Write answers in standard form. i(2+7i)(2-7i)
Verified step by step guidance1
First, recognize that the expression involves the product of three factors: \(i\), \((2 + 7i)\), and \((2 - 7i)\).
Next, multiply the two complex conjugates \((2 + 7i)\) and \((2 - 7i)\) using the difference of squares formula: \((a + bi)(a - bi) = a^2 + b^2\). So, calculate \$2^2 + 7^2$.
Simplify the result from the previous step to get a real number.
Now multiply this real number by \(i\) to get the final product.
Express the final answer in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers and Imaginary Unit
Complex numbers have a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to manipulate i is essential for working with expressions involving complex numbers.
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Introduction to Complex Numbers
Multiplication of Complex Numbers
Multiplying complex numbers involves using the distributive property (FOIL method) and combining like terms, remembering that i² = -1. This process allows simplification of products into standard form a + bi.
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Difference of Squares Formula
The expression (a + bi)(a - bi) uses the difference of squares formula: (x + y)(x - y) = x² - y². Applying this to complex conjugates simplifies the product to a real number, which is useful in simplifying complex expressions.
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