Problem 18 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ (x-8)(x+8)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 8\) and \(x = -8\).

Step 1: Understand the Equation

The given equation is \((x-8)(x+8) = 0\). This is a factored quadratic equation, and it is set to zero. We will use the zero-product property to solve it.

Step 2: Apply 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 equal to zero: \(x-8=0\) and \(x+8=0\).

Step 3: Solve Each Factor Equation Separately

First, solve \(x-8=0\). Add 8 to both sides to get \(x = 8\). Next, solve \(x+8=0\). Subtract 8 from both sides to get \(x = -8\).

Step 4: Verify Both Solutions

Substitute \(x = 8\) back into the original equation: \((8-8)(8+8) = 0\cdot16 = 0\), which is true. Then substitute \(x = -8\): \((-8-8)(-8+8) = (-16)\cdot0 = 0\), which is also true.

Key Concepts

Zero-Product Property

The Zero-Product Property is a fundamental concept in algebra that is incredibly useful when solving quadratic equations. This property states that if the product of two or more factors equals zero, at least one of the factors must be zero. This makes logical sense when you consider multiplication: anything multiplied by zero results in zero.

Applying the Zero-Product Property is a straightforward process revealed in these simple steps:

• Identify the factored form of the equation—one set to zero, like \(a \, b = 0\).
• Set each factor equal to zero, forming new simple equations like \(a=0\) and \(b=0\).
• Solve these equations separately to find possible solutions for the variable.

When you apply this property, you can efficiently find solutions for each factor, significantly simplifying the problem-solving process.

Factored Form

The factored form is a way of expressing a quadratic equation as a product of its linear factors. For the equation \(x^2 - 64 = 0\), we notice that it can be rewritten as the product \( (x-8)(x+8) = 0 \), thanks to the difference of squares identity. This immediately helps us recognize the equation is in a form that allows us to easily apply the Zero-Product Property.

Understanding when an equation is in its factored form can greatly ease the difficulty in solving it. The key is looking for expressions that fit patterns, such as:

• Difference of squares: \( a^2 - b^2 = (a-b)(a+b) \)
• Perfect square trinomials: \( (a \, b)^2 = a^2 + 2ab + b^2 \)
• Simple linear factors: \( ax + b \)

By identifying these patterns, equations can often be transformed into the factored form, simplifying the problem significantly.

Solving Equations

Solving equations is the art of finding the values of variables that satisfy the given mathematical sentences. For quadratic equations, the goal is to identify values of the variable that make the equation true.

With factored forms, solving becomes a systematic approach:

• First, identify the factors of the quadratic expression.
• Apply the Zero-Product Property to set each factor equal to zero, leading to simpler linear equations.
• Solve these linear equations individually using basic algebraic techniques like addition, subtraction, multiplication, or division.

Consider the case of \(x-8=0\) and \(x+8=0\). Solving these expressions gives us the solutions \(x=8\) and \(x=-8\). Finally, substitute these solutions back into the original equation to verify their validity. These steps guide you to consistently solve quadratic equations effectively and efficiently.