Perform the indicated operations and write the result in standard form. (−2 + √−100)²

검증된 단계별 안내

Recognize that the expression involves a complex number because of the square root of a negative number: \(\sqrt{-100}\). Recall that \(\sqrt{-1} = i\), where \(i\) is the imaginary unit.

Rewrite \(\sqrt{-100}\) as \(\sqrt{100} \times \sqrt{-1} = 10i\). So the expression becomes \((-2 + 10i)^2\).

Use the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = -2\) and \(b = 10i\).

Calculate each term separately: \(a^2 = (-2)^2\), \(2ab = 2 \times (-2) \times 10i\), and \(b^2 = (10i)^2\).

Combine the results and simplify, remembering that \(i^2 = -1\), to write the final expression in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

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주요 개념

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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 interpret and manipulate expressions involving √-100 requires recognizing that √-100 = 10i.

Operations with Complex Numbers

Performing operations like addition, multiplication, and exponentiation on complex numbers follows algebraic rules, treating i as a variable but applying i² = -1 to simplify. Squaring a complex number involves expanding the binomial and simplifying using this property.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations, the result should be simplified and expressed clearly in this form, separating the real and imaginary parts.