Evaluate x²+19 / 2−x for x = 3i.

Verified step by step guidance

1. Identify the given expression: \(\frac{x^2 + 19}{2 - x}\) and the value of \(x = 3i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
2. Substitute \(x = 3i\) into the numerator: calculate \(x^2 + 19\) by first finding \(x^2 = (3i)^2\) and then adding 19.
3. Substitute \(x = 3i\) into the denominator: calculate \$2 - x = 2 - 3i$.
4. Simplify the numerator using the fact that \(i^2 = -1\), so \((3i)^2 = 9i^2 = 9(-1) = -9\), then add 19 to get the numerator value.
5. Write the expression as a complex fraction with the simplified numerator and denominator, and if needed, multiply numerator and denominator by the conjugate of the denominator to simplify the expression further.

Key Concepts

Complex Numbers

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 substitute and manipulate complex numbers is essential when evaluating expressions involving imaginary values.

Substitution in Algebraic Expressions

Substitution involves replacing a variable in an expression with a given value. When the value is complex, careful algebraic manipulation is required to correctly simplify the expression, especially when dealing with powers and denominators.

Simplification of Rational Expressions

Simplifying rational expressions involves performing algebraic operations such as expanding, factoring, and reducing fractions. When the numerator and denominator contain complex numbers, simplification must consider the properties of complex arithmetic to reach the final simplified form.