Problem 3 - A Graphical Approach to College Algebra

Question

Match the equation in Column I with its solution(s) in Column II. Do not use a calculator. (II) A. \(\pm 2 i\), B. \(\pm 2 \sqrt{2}\), C. \(\pm i \sqrt{2}\), D. 2, E. \(\pm \sqrt{2}\), F. \(-2\), G. \(\pm 2\), H. \(\pm 2 i \sqrt{2}\) (I) $$x^{2}+2=0$$

Step-by-Step Solution

Answer

The solution is option C: \( \pm i \sqrt{2} \).

Step 1: Understanding the Equation

The given equation is \( x^2 + 2 = 0 \). This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 0 \), and \( c = 2 \).

Step 2: Move Constant to Other Side

Rewrite the equation by transferring the constant to the other side to isolate the \( x^2 \): \( x^2 = -2 \).

Step 3: Solve for x Using Square Roots

To solve for \( x \), take the square root of both sides. Since the right-hand side is negative, the solutions will be complex: \( x = \pm \sqrt{-2} \).

Step 4: Simplify the Square Root

Simplify \( \sqrt{-2} \) by expressing it as \( \sqrt{2} \times i \), where \( i \) represents the imaginary unit. Therefore, the solutions are \( x = \pm i \sqrt{2} \).

Step 5: Match the Solutions with Column II

The solutions \( \pm i \sqrt{2} \) correspond to option C in Column II.

Key Concepts

Complex Numbers

Complex numbers are a fascinating part of mathematics, expanding beyond the realm of real numbers. They are expressed in the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. This unique combination allows for the resolution of equations that have no real solutions, like \( x^2 + 2 = 0 \). In this equation, the solution involves taking the square root of a negative number, which is where complex numbers shine.

• The real part of a complex number is \( a \).
• The imaginary part is \( bi \).
• They are particularly useful in various fields such as engineering, physics, and applied mathematics.

Understanding complex numbers involves becoming comfortable with calculations involving the imaginary unit \( i \), and recognizing their critical role in extending our numerical understanding.

Imaginary Unit

The imaginary unit, denoted as \( i \), is the cornerstone of complex numbers and a crucial concept in the quadratic equation \( x^2 + 2 = 0 \). It is defined as \( i = \sqrt{-1} \). This definition allows mathematicians to extend the real number system to include solutions to equations like \( x^2 = -1 \). In the process of solving quadratic equations involving negative numbers under the square root, the imaginary unit leads to the formation of complex numbers.

• \( i^2 = -1 \) because squaring the imaginary unit returns a negative real number.
• \( i \) enables the computation of square roots of negative numbers, which are non-real numbers.
• The imaginary unit is not just theoretical; it has practical applications, especially in electrical engineering and signal processing.

When solving \( x^2 = -2 \), representing the square root of \(-2\) as \( \sqrt{2} \times i \) offers a real component for dealing with it mathematically.

Square Roots

The concept of square roots is integral to solving quadratic equations like \( x^2 + 2 = 0 \). Typically, a square root is a value that, when multiplied by itself, gives the original number. However, when dealing with negative numbers within square roots, we transition into the complex plane. Let's break it down:

• \( \sqrt{a} \) is understood as the number which, when squared, results in \( a \).
• For negative numbers, such as \( -2 \), we use the imaginary unit \( i \) to express \( \sqrt{-2} \).
• This transforms the problem from illogical in the real number system to solvable in complex numbers, yielding \( \pm i \sqrt{2} \) as solutions.

This allows mathematicians and students alike to comprehend equations with no real solutions, expanding our capabilities with the introduction of complex solutions.