Problem 16 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ a^{2}=4 $$

Step-by-Step Solution

Answer

Answer: The possible values of a are 2 and -2.

Step 1: Identify the equation

We are given the equation $$a^{2} = 4$$ and need to solve for 'a'.

Step 2: Take the square root of both sides

To extract the roots, take the square root of both sides of the equation: $$\sqrt{a^{2}} = \sqrt{4}$$.

Step 3: Simplify

Simplifying the equation, we get: $$a = \pm 2$$. So, the two possible solutions for 'a' are 2 and -2.

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions where the highest exponent of a variable is squared. These equations take the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. However, the equation in our exercise, \(a^2 = 4\), is a simpler form since it only involves squaring a single variable. Solving such equations involves finding the value of the variable that makes the equation true. These solutions are called the roots of the equation. To solve quadratic equations efficiently, various methods are used, including factoring, completing the square, using the quadratic formula, and extraction of roots. For equations that are as straightforward as \(a^2 = 4\), the extraction of roots is often the simplest method.

Extraction of Roots

The extraction of roots is a method used to solve quadratic equations that are in a specific format, such as \(x^2 = k\). Our exercise \(a^2 = 4\) falls into this category. Extracting the root involves finding two numbers that, when squared, result in the given number on the other side of the equation.

In this exercise, we began by identifying the equation \(a^2 = 4\). The main goal is to isolate the variable by applying the square root to both sides. By taking the square root of both sides, we arrive at \(\sqrt{a^2} = \sqrt{4}\). Doing so simplifies to \(a = \pm 2\). The positive and negative signs are crucial because squaring either 2 or -2 results in 4. Hence, the solutions to \(a^2 = 4\) are both 2 and -2.

This method is ideal for equations that present themselves in squared form and for which the values can be perfect squares. It provides an efficient path to finding the solution without needing to rearrange the equation extensively.

Square Roots

The concept of square roots is pivotal when solving equations by extracting roots. A square root of a number is a value that, when multiplied by itself, equals the original number. Mathematically, if \(b^2 = n\), then \(b\) is a square root of \(n\).

For the equation \(a^2 = 4\), we find the square root of 4, which is 2. However, it isn't just 2; it is \(\pm 2\), because both positive and negative values satisfy the condition when squared. The symbol \(\pm\) denotes both possibilities.

Recognizing that both a positive and a negative root exist is an essential aspect of correctly solving such equations. This dual solution arises because both "+a" and "-a" yield the same square when squared. In practical scenarios, knowing about square roots broadens the understanding of quadratic solutions, beneficial in various academic and real-world applications.