Problem 36 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ x^{2}-4=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are x = 2 and x = -2.

Step 1: Write the equation in the standard form

Let's write the given equation in the standard quadratic form as: $$x^2 - 4 = 0$$

Step 2: Factor the quadratic expression

The difference of squares can be factored into two binomials as: $$(x-2)(x+2)=0$$

Step 3: Solve for x using the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero and solve for x: $$x - 2 = 0 \Rightarrow x = 2$$ $$x + 2 = 0 \Rightarrow x = -2$$

Step 4: Write the final solution

The solutions of the quadratic equation, $$x^2-4=0$$, are: $$x = 2 \text{ and } x = -2$$

Key Concepts

Understanding Factoring in Quadratic Equations

Factoring is a method used to simplify quadratic equations into a set of simple expressions that can be easily solved. When you factor an equation, you're essentially breaking it down into its components or ‘factors’ that, when multiplied together, yield the original equation.
In the context of our quadratic equation, \(x^2 - 4 = 0\), we can recognize a special pattern called a 'difference of squares'.
This allows us to express the equation as the product of two binomials: \((x-2)(x+2)\).

• The expression \(x^2 - 4\) is split into \((x-2)\) and \((x+2)\).
• Notice how each factor contains an \(x\) term and a constant, which, when multiplied together, return to the original quadratic expression.

Factoring makes it easier to apply the Zero Product Property to find the solutions for \(x\). This method is highly useful, especially for algebra students learning to solve quadratic equations.

Applying the Zero Product Property

The Zero Product Property is a fundamental principle in algebra that simplifies the process of solving equations like our quadratic equation.
This property states that if the product of two numbers is zero, then at least one of the numbers must also be zero.
For our factored equation \((x-2)(x+2)=0\), we apply this principle as follows:

• Set each factor equal to zero: \(x-2=0\) and \(x+2=0\).
• Solving these two simple equations, we find that \(x = 2\) and \(x = -2\).

These are the solutions for the quadratic equation \(x^2 - 4 = 0\). This property is a powerful tool because it reduces potentially complicated equations into straightforward arithmetic operations, making the solving process more intuitive.

Recognizing and Using the Difference of Squares

The difference of squares is a special pattern in algebra that occurs when we subtract one squared number from another: \(a^2 - b^2\).
This pattern can be expressed as the product of two binomials: \((a-b)(a+b)\).
Using this pattern, our original quadratic equation \(x^2 - 4 = 0\) can be rewritten.

• Here, \(a^2\) is \(x^2\), and \(b^2\) is \(4\) or \(2^2\).
• Using the difference of squares formula, we rewrite it as \((x-2)(x+2)\).

Recognizing this pattern can greatly simplify solving quadratic equations. It allows us to almost immediately convert the expression to a factored form, paving the way for the application of the Zero Product Property. It's a neat shortcut that can save time and effort in algebraic manipulations.