Problem 1 - 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 solutions to the equation \(x^2 = 4\) are \(\pm 2\), which matches option G.

Step 1: Understand the Equation

We are given the equation \( x^2 = 4 \) and we need to solve for \( x \). This means we are looking for all values of \( x \) that satisfy this equation.

Step 2: Apply Square Roots

To solve \( x^2 = 4 \), we take the square root of both sides of the equation. So, we have \( x = \pm \sqrt{4} \).

Step 3: Simplify the Square Root

Simplify \( \sqrt{4} \) to get \( 2 \). Thus, the solutions are \( x = \pm 2 \).

Step 4: Match with Options

According to Column II, option G is \( \pm 2 \). Hence, \( x = \pm 2 \) matches with option G.

Key Concepts

Square Roots

The square root is a fundamental mathematical operation that simplifies the square of a number. It is the inverse process of squaring. For a non-negative number, say 4, the square root is another non-negative number which, when multiplied by itself, gives the original number back. In our exercise, the given equation is \( x^2 = 4 \). To isolate \( x \), we take the square root of both sides:
\[ x = \pm \sqrt{4} \]
This operation yields two results: 2 and -2, owing to the fact that both \( (2)^2 \) and \( (-2)^2 \) equal 4. It is important to remember that the positive and negative roots are considered because squaring either results in the same positive number. Square roots are essential not only in algebra but also in geometry and calculus.

• Positive root: Gives the principal (non-negative) solution.
• Negative root: Reflects the symmetry in the squared term with respect to zero.

Learning to find square roots accurately is a skill that underpins much of algebraic problem-solving.

Equation Solving

Equation solving is the art of finding all values of the variable that turn the equation into a true statement. With quadratic equations, such as \( x^2 = 4 \), we employ different techniques depending on the complexity of the equation. Solving quadratic equations might involve factoring, using the quadratic formula, or taking square roots. In our particular instance, solving \( x^2 = 4 \) is straightforward because it can be directly simplified by taking the square root of both sides:
\[ x = \pm \sqrt{4} \]
This directly provides the solutions \( x = 2 \) and \( x = -2 \).

• Step 1: Identify the equation structure (is it quadratic, linear, etc.?).
• Step 2: Choose the appropriate method (in this case, square roots) to solve it.
• Step 3: Simplify the equation and find the solutions.

Keep in mind that for equations involving square roots, checking the solutions by substituting back into the original equation is a good practice to ensure accuracy.

Imaginary Numbers

Imaginary numbers extend the real number line to cover the square roots of negative numbers, introducing the unit \( i \), where \( i^2 = -1 \). This concept is important for understanding solutions to equations that yield no real number result when simplified. If the given equation were \( x^2 = -4 \) instead, conventional methods would not suffice, as \( x^2 \) cannot equal a negative number with real solutions.

To solve such a quadratic equation, we employ imaginary numbers:
\[ x = \pm \sqrt{-4} = \pm 2i \]
Here, \( \pm 2i \) are the solutions in the complex plane. Imaginary numbers enable us to solve problems in electrical engineering, signal processing, and dynamic systems, among others.

• Real part: Coefficient of 1 in conventional numbers.
• Imaginary part: Expressed as a multiple of \( i \).
• Complex solutions: The combination of real and imaginary parts.

Introducing imaginary numbers expands our mathematical toolkit, allowing solutions that wouldn't be possible with real numbers alone.