Problem 4 - 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 \( E. \\pm \sqrt{2} \).

Step 1: Understand the given equation

The equation provided is a quadratic equation: \(x^2 - 2 = 0\). Our task is to find the values of \(x\) which make this equation true.

Step 2: Isolate the quadratic term

Rearrange the equation \(x^2 - 2 = 0\) to isolate the quadratic term by adding 2 on both sides. This results in \(x^2 = 2\).

Step 3: Solve for x using square roots

To solve for \(x\), take the square root of both sides of the equation \(x^2 = 2\). This yields two potential solutions due to the nature of square roots: \(x = \pm \sqrt{2}\).

Step 4: Match solution with options

Look at Column II to find the matching solution for \(x = \pm \sqrt{2}\). It corresponds to option \(E. \pm \sqrt{2}\).

Key Concepts

Square Roots

Taking the square root is a central process in solving quadratic equations. When you see an equation like \(x^2 = 2\), the goal is to find out what value of \(x\) will make the equation true.
This involves determining the square root of both sides of the equation. However, it’s important to remember that square roots can have two outcomes.
When dealing with square roots, always keep in mind:

• Every positive number has two square roots: one positive and one negative.
• The operation allows us to work through squares to find their base value.

For example, with \(x^2 = 2\), taking the square root of both sides gives \(x = \pm \sqrt{2}\).
This means that both \(\sqrt{2}\) and \(-\sqrt{2}\) can satisfy our original equation, reflecting the two possible roots of the original equation.
Square roots of negative numbers become a bit more complex, which leads us into complex numbers—an exciting topic of its own.

Solving Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\).
In our example, the equation is \(x^2 - 2 = 0\), which is a simpler form where \(a = 1\), \(b = 0\), and \(c = -2\).
Here's how to tackle these equations:

• Isolate the quadratic part: Begin by moving terms around so that the equation is in its simplest form, \(x^2 = 2\).
• Use square roots: Solve for \(x\) by taking the square root of both sides, remembering to consider both positive and negative roots, \(x = \pm \sqrt{2}\).

Many students find it useful to visualize this process as balancing an equation: what you do on one side, you must do on the other.
This method is sometimes referred to as the "square root method," one of several ways to solve quadratics, others being factoring and using the quadratic formula.
Always match your solution back to the original question options to ensure you have the correct answer.

Complex Numbers

Complex numbers introduce a system for solving quadratic equations where the roots are not simply real numbers.
If you encounter an equation resulting in the need for the square root of a negative number, then complex numbers provide a solution.

• Imaginary Unit: The basic element of complex numbers is \(i\), defined as the square root of \(-1\). This allows us to resolve expressions like \(\sqrt{-4}\) as \(2i\).
• Form of Complex Numbers: Complex numbers are typically expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

In complex-oriented equations, if you find yourself needing to take the square root of a negative number, you’ll inevitably deal with \(i\).
For instance, an equation like \(x^2 + 4 = 0\) leads us to solve \(x^2 = -4\).
Taking square roots results in \(x = \pm 2i\).
Understanding complex numbers is vital, as they expand the framework of what solutions look like beyond the limitations of real-only numbers, providing an essential tool in algebra and beyond.