Textbook Question
Work each problem. Show that -2+i is a solution of the equation x²+4x+5=0.
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1. Equations & Inequalities
Complex Numbers
3:04 minutesProblem 27
Textbook Question
Perform each operation. Write answers in standard form. -5i(3-i)²
Verified step by step guidance1
First, recognize that the problem requires you to simplify the expression \(-5i(3 - i)^2\) and write the answer in standard form, which is \(a + bi\) where \(a\) and \(b\) are real numbers.
Start by expanding the squared term \((3 - i)^2\). Use the formula for squaring a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 3\) and \(b = i\), so write out the expansion as \(3^2 - 2 \times 3 \times i + i^2\).
Calculate each part of the expansion: \$3^2 = 9\(, \(-2 \times 3 \times i = -6i\), and recall that \)i^2 = -1\(. Substitute these values back into the expression to get \)9 - 6i + (-1)$.
Combine the real terms \$9\( and \)-1\( to simplify the expression inside the parentheses to \)8 - 6i\(. Now, multiply this result by \)-5i\(, so you have \)-5i(8 - 6i)$.
Distribute \(-5i\) across both terms inside the parentheses: multiply \(-5i \times 8\) and \(-5i \times (-6i)\). Remember to use the property \(i^2 = -1\) when simplifying the product involving \(i^2\). After this, combine like terms to write the expression in the form \(a + bi\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers and Imaginary Unit
Complex numbers consist of 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 simplifying expressions involving imaginary numbers.
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Introduction to Complex Numbers
Exponentiation of Binomials
Raising a binomial to a power, such as (3 - i)², involves applying the distributive property or the formula (a - b)² = a² - 2ab + b². This step is crucial for expanding and simplifying the expression before further operations.
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Multiplication of Complex Numbers
Multiplying complex numbers requires distributing each term and combining like terms, remembering to replace i² with -1. This process helps in simplifying the product into standard form a + bi.
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