Problem 37 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possib $$4 x^{2}-16=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = 2\).

Step 1: Transform the Equation

The given equation is \(4x^2 - 16 = 0\). Let's transform it into a more easily solvable form by factorizing. This is possible because \(4x^2 - 16 = 0\) is a difference of squares, which can be written as \((2x)^2 - 4^2 = 0\), that further simplifies to \((2x + 4)(2x - 4) = 0\).

Step 2: Solve Each Factor

Set each factor to zero and solve for x: 1. \(2x + 4 = 0\), then \(2x = -4\), finally \(x = -2\). 2. \(2x - 4 = 0\), then \(2x = 4\), finally \(x = 2\). So the solution to the equation are \(x = -2\) and \(x = 2\).

Key Concepts

Difference of Squares

Understanding the difference of squares is key to solving quadratic equations like the one in this example. The expression \(a^2 - b^2\) is known as a difference of squares because it involves the subtraction of one square term from another. This expression can always be factored into \((a+b)(a-b)\). The concept is widely used in algebra to simplify and solve equations. In the given exercise, \(4x^2 - 16 = 0\), it can be seen as a difference of squares:

• The term \(4x^2\) is \((2x)^2\).
• Similarly, the term \(16\) is \(4^2\).

Recognizing these squares allows us to transform \(4x^2 - 16\) into \((2x + 4)(2x - 4) = 0\). This transformation makes the equation easier to solve by breaking it into simpler factors.

Factorization Method

Factorization is a technique used to solve quadratic equations by expressing the equation in terms of its factors. The factorization method involves writing a polynomial as the product of its factors. This method is efficient and frequently used due to its straightforward approach. In our example, the equation \((2x + 4)(2x - 4) = 0\) was derived using factorization.

• This equation is already factored into two binomials.
• We then take each of these factors and set them equal to zero.

The factorization method turns a more complex expression into simpler ones that are easier to solve. For the factors \(2x + 4 = 0\) and \(2x - 4 = 0\), solving each provides the roots of the original equation. This method not only facilitates solving the equation but also reveals the solutions in a very structured and logical way.

Equation Simplification

Equation simplification makes solving complex expressions more manageable. To simplify an equation means transforming it into a form that is easier to handle. This usually involves factoring, combining like terms, or other algebraic manipulations. In the context of the equation \(4x^2 - 16 = 0\), simplification was achieved through the difference of squares.

• First, identify parts of the equation that can be simplified, such as difference of squares.
• Then, rewrite these parts using algebraic identities, making the equation easier to work with.

In our exercise, simplification first turned \(4x^2 - 16\) into \((2x)^2 - 4^2\) and then into \((2x+4)(2x-4)=0\). Finally, by solving each resulting simpler factor, we find the solution. This process turns a potentially difficult problem into a straightforward solution.