Problem 2 - 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) $$x^{2}=-4$$

Step-by-Step Solution

Answer

The solution is A: \( \pm 2i \).

Step 1: Identify the Type of Equation

The given equation is \( x^2 = -4 \). This is a quadratic equation where the right side is negative, which implies that the solutions are complex numbers.

Step 2: Rewrite Equation in Standard Form

Rewrite the equation in standard form: \( x^2 + 4 = 0 \). This helps identify that the sum of the roots squared equals zero because the constant \( 4 \) needs to counterbalance the left side.

Step 3: Solve for x Using Square Root Method

To solve \( x^2 = -4 \), take the square root of both sides: \( x = \pm \sqrt{-4} \). Since \( \sqrt{-4} \) involves a negative under the square root, it requires the use of imaginary numbers.

Step 4: Simplify the Radical

Recognize that \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \). Thus, \( x = \pm 2i \).

Step 5: Match with the Solutions

Compare \( \pm 2i \) with the options given in Column II. The solutions match with option A: \( \pm 2i \).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, usually written in the form \( ax^2 + bx + c = 0 \). They appear often in algebra and are fundamental in understanding how we model different types of problems. The general components are the quadratic term \( ax^2 \), the linear term \( bx \), and the constant term \( c \).

To solve these equations, several methods can be used, such as:

• Factoring: breaking down the quadratic into products of simpler expressions.
• Completing the square: redistributing values to form a perfect square trinomial.
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides solutions directly.
• Square Root Method: used when the equation can be arranged to resemble \( x^2 = k \).

Quadratic equations can have real or complex solutions, depending on the discriminant \( b^2 - 4ac \). If the discriminant is negative, as in our original exercise, the solutions are complex.

Imaginary Numbers

Imaginary numbers arise when we try to take the square root of negative numbers. In mathematics, this involves the imaginary unit \( i \), defined as \( i = \sqrt{-1} \).

They were introduced to extend the real number system and solve equations that don’t have real solutions.
They have properties such as:

• \( i^2 = -1 \)
• \( i^3 = -i \)
• \( i^4 = 1 \)

In our original exercise, the equation \( x^2 = -4 \) leads to the solution \( x = \pm 2i \).
This means that the equation doesn't have real solutions but instead has purely imaginary solutions, demonstrating the usefulness of imaginary numbers in solving quadratic equations with negative discriminants.

Square Root Method

The square root method is a straightforward approach to solving quadratic equations that can be written in terms of a square. It's particularly useful for equations in the form \( x^2 = k \).

To use this method, follow these easy steps:

• Isolate the square on one side, if necessary, to get \( x^2 = k \).
• Take the square root of both sides of the equation: \( x = \pm \sqrt{k} \).

When \( k \) is negative, such as in \( x^2 = -4 \), the square root involves imaginary numbers since a real number cannot have a negative square. This results in solutions like \( x = \pm 2i \), as calculated in the original exercise. This method efficiently handles both real and complex solutions without the need for more complex techniques.