Problem 25 - Elementary and Intermediate Algebra

Question

Consecutive Integer Problems Customer Service. At a pharmacy, customers take a ticket to reserve their turn for service. If the product of the ticket number now being served and the next ticket number to be served is \(156,\) what number is now being served? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The ticket number now being served is 12.

Step 1: Set up the variables

Let the current ticket number being served be denoted by \( x \). The next ticket number that will be served is \( x + 1 \).

Step 2: Formulate the equation

The problem states that the product of the current ticket number \( x \) and the next ticket number \( x + 1 \) equals 156. Therefore, we write the equation as:\[x(x + 1) = 156\]

Step 3: Expand the equation

Expanding the equation from Step 2 gives:\[x^2 + x = 156\]

Step 4: Rearrange into standard quadratic form

Move everything to one side of the equation to get:\[x^2 + x - 156 = 0\]

Step 5: Solve the quadratic equation

We need to find two numbers that multiply to \(-156\) and add to \(1\). These numbers are \(13\) and \(-12\). So the quadratic can be factored as:\[(x - 12)(x + 13) = 0\]

Step 6: Find the solutions for \( x \)

Set the factors equal to zero to find the solutions:1. \(x - 12 = 0\) gives \(x = 12\)2. \(x + 13 = 0\) gives \(x = -13\) (which doesn't make sense in this context as ticket numbers must be positive).

Step 7: Interpret the result

Since ticket numbers cannot be negative, the only valid solution is \(x = 12\). This means ticket number 12 is now being served.

Key Concepts

Quadratic Equations

Quadratic equations are significant in algebra and appear in lots of different problems, such as this consecutive integer issue. A basic quadratic equation is usually expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. This type of problem requires us to find the value of \(x\) that makes the equation true.

Here, we derived our equation from the given conditions: the product of two consecutive integer ticket numbers is 156. By translating words into mathematics, we form the quadratic equation \(x(x + 1) = 156\). Expanding and simplifying it leads us to \(x^2 + x - 156 = 0\).

• The highest power of the variable \(x\) is squared, hence the name quadratic.
• Solving quadratics often requires rearranging terms so that one side equals zero.

Understanding and formulating the equation in this way is crucial for solving the problem correctly.

Factoring

Factoring is a critical technique in algebra that lets us rewrite quadratics as the product of simpler expressions. This process simplifies finding solutions to equations. Once we've arranged the quadratic in standard form \(x^2 + x - 156 = 0\), we seek two numbers that multiply to \(-156\) and sum to \(1\). These numbers are \(-12\) and \(13\).

The equation splits into two factors, \((x-12)(x+13) = 0\). By setting each factor equal to zero, we can solve for \(x\); namely, \(x - 12 = 0\) and \(x + 13 = 0\).

• Each factor represents a possible solution.
• Ensuring the context matches, we discard any solution that doesn’t fit the situation—like negative ticket numbers.

The facet of factoring simplifies complex quadratic equations and makes solving them a much easier task.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations that represent a specific value or condition. In the problem at hand, the expression \(x(x + 1)\) signifies the product of two consecutive whole numbers, which is foundational to forming our initial equation.

Understanding these expressions allows us to manipulate and solve equations more effectively. Here's a simple breakdown of managing algebraic expressions:

• Recognize patterns or known identities, such as \(a(a+b)\).
• Simplify expressions by expanding or factoring where possible.
• Re-arrange terms to make solving more straightforward, typically setting the equation to zero, as we've done here.

Just like a mathematical sentence, algebraic expressions speak about relationships and quantities, giving meaning and solvability to real-world problems.