Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$(x+3)(x-5)=x^{2}-2(x+7)$$

Step-by-Step Solution

Answer

The given equation is a conditional equation, since it is only true for x = 1.

Step 1: Expand the left side of the equation

By distributing the terms, \( (x+3)(x-5) \) is expanded to \( x^{2} - 2x -15 \)

Step 2: Simplify the right side of the equation

By distributing the -2, \( x^{2}-2(x+7) \) simplifies to \( x^{2}-2x-14 \)

Step 3: Compare the two sides

At this point, we can see that the left side and right side are not equivalent. However, there may be some x-values that could make them equal - this is a hint that this is likely to be a conditional equation.

Step 4: Set the two expressions equal to solve for x

Setting \( x^{2} - 2x -15 = x^{2}-2x-14 \) and solving for x, we find that x = 1. This means that the equation is only true when x = 1. As such, this is a conditional equation as it is only true for this particular value of x.

Key Concepts

Identity

In mathematics, an equation is called an *identity* when it is true for all values of the variables involved. This means that no matter what value you substitute into the equation, it will always hold true. This type of equation reflects a universal truth, like a law that is always constant. An example of an identity is the equation

• \((x + 3)(x - 3) = x^2 - 9\)

Here, no matter what value you choose for \(x\), both sides of the equation remain equal. In the exercise given, the original equation:\[(x+3)(x-5) = x^2 - 2(x + 7)\]did not qualify as an identity because it is not valid for each value of \(x\), but only for a specific one, which we'll discuss later. If ever you find an equation that is always true, you can confidently label it as an identity.

Contradiction

A *contradiction* occurs in mathematics when an equation is never true, regardless of the value substituted into it. It means that no possible value of the variable can ever satisfy the equation. An example of a contradiction is:

• \(x + 2 = x + 5\)

In this case, no value for \(x\) will make the two sides equal, as they imply different sums regardless of \(x\). In our previous exercise, the equation is not a contradiction because it is possible to find a value for \(x\) that makes both sides equal, as shown in the solution where \(x = 1\) makes the equation true. If no such value exists, then you have a contradiction.

Expansion and Simplification

In algebra, *expansion* and *simplification* are crucial processes that help in solving equations. **Expansion** involves multiplying expressions to remove parentheses, making equations easier to manage. For the given exercise, we started with expanding the left side:

• The expression \((x+3)(x-5)\) was expanded to \(x^2 - 2x - 15\)

This process involves distributing each term in the first parenthesis with each in the second.
**Simplification** refers to reducing an expression to its simplest form. This can involve combining like terms or distributing factors like the -2 in the right side of the equation:

• \(x^2 - 2(x + 7)\) simplifies to \(x^2 - 2x - 14\)

After expanding and simplifying both sides of the equation, it becomes clear whether the two sides can be equated for some value of \(x\), helping identify its nature. In our exercise, expansion and simplification showed us that the equation is only valid under specific conditions, leading to it being categorized as a conditional equation.