Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the algebraic expressions. If \(y=2 x^{2}+x-3,\) evaluate \(y\) given \(x=2-3 i\).

Step-by-Step Solution

Answer

The expression evaluates to \(y = -11 - 27i\).

Step 1: Substitute the Given Value

Start by substituting the value of \(x\) into the expression \(y = 2x^2 + x - 3\). The given value of \(x\) is \(2 - 3i\). Substitute \(x = 2 - 3i\) into the expression: \[ y = 2(2 - 3i)^2 + (2 - 3i) - 3 \]

Step 2: Expand the Squared Term

Calculate \((2 - 3i)^2\) by using the formula \((a - b)^2 = a^2 - 2ab + b^2\). Substitute \(a = 2\) and \(b = 3i\): \((2 - 3i)^2 = 2^2 - 2(2)(3i) + (3i)^2 = 4 - 12i + 9(-1)\)Remember that \(i^2 = -1\). Simplifying gives: \((2 - 3i)^2 = 4 - 12i - 9 = -5 - 12i\).

Step 3: Multiply by 2

Now multiply the result by 2: \[2(-5 - 12i) = -10 - 24i\]

Step 4: Simplify the Expression

Substitute the simplified values back into the expression from Step 1: \[ y = -10 - 24i + (2 - 3i) - 3\]Combine the real and imaginary parts separately:\[ y = (-10 + 2 - 3) + (-24i - 3i)\] Compute the sums: \[ y = -11 - 27i\]

Step 5: Final Result

The evaluated expression for \(y\) when \(x = 2 - 3i\) is \[y = -11 - 27i\].

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations such as addition, subtraction, multiplication, and division. In this exercise, we look at the expression \( y = 2x^2 + x - 3 \). This expression contains:

• Coefficients, such as 2 and 1
• A variable \( x \)
• Constants, such as -3

The goal is to substitute a specific value for \( x \) to find the corresponding value of \( y \). Evaluating algebraic expressions involves replacing variables with given values and performing the operations indicated. It's crucial to follow the order of operations - parentheses first, exponents second, then multiplication and division from left to right, and finally, addition and subtraction from left to right.

Understanding algebraic expressions and their components is fundamental to solving many types of mathematical problems, including those involving complex numbers.

Substitution Method

The substitution method is a straightforward strategy used in algebra to replace one variable with a given value in an expression or equation. In this case, we're replacing \( x \) with the complex number \( 2 - 3i \).

To apply the substitution method:

• Identify the variable in the expression that needs to be replaced or evaluated.
• Check the given values for these variables.
• Substitute each occurrence of the variable with the given value in the expression.

After substituting \( x = 2 - 3i \) into the expression \( y = 2x^2 + x - 3 \), calculations are performed to evaluate \( y \). This approach helps in simplifying expressions and finding specific solutions to problems quickly by eliminating the variable and working exclusively with numbers.

Imaginary Unit

The imaginary unit, denoted by \( i \), is a fundamental concept in complex numbers. It is defined as \( i = \sqrt{-1} \), meaning \( i^2 = -1 \). This property allows us to work with square roots of negative numbers by extending our number system to include imaginary numbers.
In this exercise, complex numbers are involved, and the expression \( x = 2 - 3i \) contains the imaginary unit. When you multiply or square complex numbers, remember:

• \( i^2 = -1 \) - This simplification is essential in any calculation.
• Complex multiplication involves combining real and imaginary components separately.

For instance, when calculating \((2 - 3i)^2\), the \(i^2\) results in \(-1\), affecting both the real and imaginary parts of the solution. Understanding the role and behavior of the imaginary unit is crucial not only in complex number calculations but also in solving algebraic expressions involving them. By mastering operations with \( i \), you will find it easier to tackle complex mathematical challenges.