Problem 9 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$x^{2}-4=0$$

Step-by-Step Solution

Answer

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

Step 1: Set Equation to Zero

The given equation is already set to zero: \(x^2 - 4 = 0\). This is the correct form to start solving by factoring.

Step 2: Identify the Difference of Squares

Notice that \(x^2 - 4\) is a difference of squares, which is a special factoring case. It can be rewritten using the formula \(a^2 - b^2 = (a+b)(a-b)\).

Step 3: Factor the Equation

Here, \(x^2\) is \(a^2\) and \(4\) is \(2^2\), so you can write it as \((x + 2)(x - 2) = 0\).

Step 4: Solve for Roots

To solve for \(x\), set each factor equal to zero: \(x + 2 = 0\) or \(x - 2 = 0\).

Step 5: Find the Solutions

Solving \(x + 2 = 0\) gives \(x = -2\) and solving \(x - 2 = 0\) gives \(x = 2\). Both are solutions to the equation.

Key Concepts

Factoring

Factoring is a technique used in mathematics to break down complex expressions into simpler ones. This is often the first step in solving quadratic equations. The idea is to express a quadratic in the form

• \( ax^2 + bx + c = 0 \)

as the product of two binomials. When an equation can be factored, it means you can find two numbers that multiply to give the constant term \( c \) while also adding to give the linear coefficient \( b \).
This method works when the equation is set to zero, making it easier to find solutions for \( x \). Factoring simplifies a quadratic equation so that each factor can be set separately equal to zero, unveiling the values of \( x \) that satisfy the equation. If you encounter a situation where factoring seems impossible, other methods such as completing the square or using the quadratic formula might be necessary.

Difference of Squares

The difference of squares is a special pattern that occurs in algebra where an expression is structured as \( a^2 - b^2 \). This can be factored into:

• \( (a+b)(a-b) \)

This pattern is useful for simplifying and solving quadratic equations.
In the exercise provided, the expression \( x^2 - 4 \) is a straightforward example of the difference of squares with \( a = x \) and \( b = 2 \). Recognizing such patterns makes it quick to factor and solve equations. Using this pattern significantly reduces the complexity and time it takes to solve quadratic equations as you can immediately rewrite the equation in its factored form. Always look for opportunities to apply this rule to save time during your calculations.

Solving Equations

Solving quadratic equations involves finding the value of the variable that makes the equation true. After factoring a quadratic equation, you'll typically end up with two binomials set equal to zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero.
Therefore, set each binomial to zero separately and solve each resulting equation:

• \( a+b = 0 \)
• \( a-b = 0 \)

This process will yield the solutions for the variable. In the example \((x + 2)(x - 2) = 0\), setting each part of the product to zero gives the individual equations \(x + 2 = 0\) and \(x - 2 = 0\). Solving each gives the roots \(x = -2\) and \(x = 2\). This method leverages simple arithmetic operations to isolate the variable and provides a straightforward path to solutions.

Algebra

Algebra is like the language of mathematics, where symbols stand in for numbers or specific values. This allows for the manipulation of mathematical sentences. The goal in algebra often involves solving for an unknown variable through various operations.
Quadratic equations are a common area in algebra where you seek to find the roots or solutions of the equation. Familiarity with different algebraic techniques such as factoring, difference of squares, and others is crucial for efficiently solving these equations. Algebraic expressions and equations are foundational to not just mathematics but numerous fields that apply logical problem-solving to real-world situations.
Understanding these skills is vital as they serve as building blocks for further study in higher mathematics, science, finance, and engineering.