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Chapter 4: Problem 91

Solve each equation. $$(x+4)(x-4)=(x-2)(x+6)$$

Short Answer

Expert verified
The solution to the equation is \(x=-1\).

Step by step solution

01

Expand all binomials

To begin, we expand both sides of the equation using the formula \((a-b)(a+b)=a^2-b^2\). The equation becomes: \(x^2 - 16 = x^2+4x-12\)
02

Simplifying the equation

Next, simplify the equation by removing the like terms on both sides. This leads to: \(-16 = 4x - 12\)
03

Solve for x

Finally we can solve the equation for x by adding 12 to both sides and then dividing by 4. This gets us: \(x = -1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion
When you come across expressions like \((x + 4)(x - 4)\), you can expand them using the concept of binomial expansion. This process involves multiplying each term in one binomial by each term in the other. For the expression \((x + 4)(x - 4)\), the result is actually a special product known as the difference of squares. The difference of squares is a shortcut formula \((a - b)(a + b) = a^2 - b^2\). Therefore:
  • \(a = x\) and \(b = 4\),
  • resulting in \(x^2 - 16\).
This expansion method simplifies the equation, making it easier to solve. Mastering binomial expansion will help you solve complex problems more efficiently.
Simplifying Algebraic Expressions
Once you have expanded the binomials, the next step is simplifying the algebraic expressions. Simplification involves combining like terms and reducing the expression to its simplest form. This process makes the equation much easier to handle. In the given equation:
  • After expanding, we have the equation \(x^2 - 16 = x^2 + 4x - 12\).
  • Notice that both sides contain the term \(x^2\). By removing these common terms, the equation simplifies to \(-16 = 4x - 12\).
This process clears out the redundant parts and sets up a cleaner path for solving the equation. Simplification is key to minimizing errors and reaching the solution accurately.
Solving Linear Equations
The final part is solving the linear equation that results from our simplification efforts. A linear equation in one variable is simpler compared to quadratic equations. The goal is to isolate the variable, typically \(x\), on one side of the equation. With the simplified equation:
  • \(-16 = 4x - 12\)
  • First, add 12 to both sides to obtain: \(-4 = 4x\).
  • Then, divide by 4 to isolate \(x\), giving us: \(x = -1\).
Solving linear equations often involves basic operations: addition, subtraction, multiplication, or division. These steps highlight the importance of performing the same operation on both sides to maintain the equality. Mastering these concepts is crucial for handling equations systematically and arriving at accurate solutions.

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