Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the algebraic expressions. If \(y=x^{2}+x-4,\) evaluate \(y\) given \(x=2 i\).

Step-by-Step Solution

Answer

The evaluated expression is \( y = -8 + 2i \).

Step 1: Identify the Expression

The expression given is \( y = x^2 + x - 4 \). We need to evaluate this expression for \( x = 2i \).

Step 2: Substitute the Value of x

Substitute \( x = 2i \) into the expression so that it becomes \( y = (2i)^2 + 2i - 4 \).

Step 3: Calculate the Square of 2i

Calculate \( (2i)^2 \). We know that \( i^2 = -1 \), so \( (2i)^2 = 4i^2 = 4(-1) = -4 \).

Step 4: Substitute and Simplify the Expression

Substitute \( -4 \) from Step 3 back into the expression: \( y = -4 + 2i - 4 \). Simplify it further to get \( y = -8 + 2i \).

Step 5: Write the Final Answer

The expression simplifies to \( y = -8 + 2i \). Thus, when \( x = 2i \), \( y = -8 + 2i \).

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, providing a way to solve equations that doesn't work in the real number system. At their core, complex numbers consist of two parts: a real part and an imaginary part. A typical complex number is expressed as \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit, which satisfies the equation \(i^2 = -1\).

One of the benefits of using complex numbers is their ability to handle square roots of negative numbers. In the real number system, these do not have solutions, but in the complex number system, they do. For example, the solution for the square root of -4 would be \(2i\).

In calculations, like those in our original problem, it's important to apply the properties of complex numbers correctly, especially when multiplying the imaginary unit. Understanding that \(i^2 = -1\) helps simplify expressions and solve equations involving complex numbers.

Substitution Method

The substitution method is a valuable technique in algebra that allows us to replace a variable with a specific value, simplifying the evaluation of expressions. It's an essential skill in complex calculations and can turn complex problems into more manageable tasks.

This method involves a simple process:

• Identify the variable to substitute.
• Replace every instance of this variable in your equation or expression with the given value.
• Perform any necessary calculations to simplify the expression further.

In the original exercise, for instance, we have the algebraic expression \(y = x^2 + x - 4\) that needs evaluation with \(x = 2i\). By substituting the complex number \(2i\) for \(x\), we transform the problem into a straightforward sequence of arithmetic operations. This makes analysis simpler and helps us to quickly find the final solution, \(y = -8 + 2i\).

Using substitution not only aids in solving expressions but is also fundamental in solving systems of equations and other algebraic manipulations.

Simplification in Algebra

Algebra is all about finding simplicity in mathematical expressions. Once you substitute variables with numbers, the next crucial step is simplification. In the realm of algebra, simplification aims to make expressions as compact as possible while retaining their meaning and properties.

Simplification involves several operations:

• Combining like terms: Terms that have the same variables and powers are added or subtracted together, e.g., \(2x + 3x = 5x\).
• Applying arithmetic operations carefully, especially with complex numbers, as they follow unique rules.
• Utilizing the properties of arithmetic: such as distributive, associative, and commutative properties to rearrange and reduce expressions.

In our exercise, simplifying \((2i)^2\) to \(-4\) was a crucial simplification step. Likewise, combining \(-4 + 2i - 4\) to give the final simplified answer of \(y = -8 + 2i\), made the expression clearer and resolved.

Simplification isn’t just a procedure; it’s a mindset. By always seeking the simplest form of an expression, you're also gaining deeper insights into its structure and behavior, paving the way for more advanced mathematical exploration.