Problem 42 - Elementary and Intermediate Algebra

Question

Quadratic Equation Model Problems Crafts. The illustration shows how a wall hanging can be created by stretching yarn from peg to peg across a wooden ring. The relationship between the number of pegs \(p\) placed evenly around the ring and the number of yarn segments \(s\) that criss-cross the ring is given by the formula \(s=\frac{p(p-3)}{2} .\) How many pegs are needed if the designer wants 27 segments to criss-cross the ring? (Hint: Multiply both sides of the equation by \(2 .\) )

Step-by-Step Solution

Answer

The designer needs 9 pegs.

Step 1: Convert the Word Problem to an Equation

We are given the formula that relates the number of pegs, \( p \), to the number of yarn segments, \( s \):\[s = \frac{p(p - 3)}{2}\]We want to find \( p \) when \( s = 27 \). Substitute \( 27 \) for \( s \) in the formula:\[27 = \frac{p(p - 3)}{2}\]

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by 2:\[2 \times 27 = p(p - 3)\]This simplifies to:\[54 = p^2 - 3p\]

Step 3: Set Up Quadratic Equation

Rearrange the equation to set it to zero:\[p^2 - 3p - 54 = 0\]This is now a standard quadratic equation in the form \( ax^2 + bx + c = 0 \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -3 \), and \( c = -54 \).Calculate the discriminant:\[b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-54)\]\[= 9 + 216\]\[= 225\]Use the quadratic formula:\[p = \frac{-(-3) \pm \sqrt{225}}{2 \cdot 1}\]\[p = \frac{3 \pm 15}{2}\]Determine possible values for \( p \):\[p = \frac{3 + 15}{2} = 9\]\[p = \frac{3 - 15}{2} = -6\]

Step 5: Choose the Valid Solution

The number of pegs \( p \) must be a positive integer, so \( p = -6 \) is not possible. Thus, the valid solution is \( p = 9 \).

Key Concepts

Algebraic Problem Solving

Algebraic problem solving involves taking a real-world situation and translating it into a mathematical equation. This often requires you to understand the relationships between different elements in a problem. Let's break it down using the wall hanging example.
In this situation, we know the number of yarn segments depends on the number of pegs placed evenly around a wooden ring. The formula is given as: \( s = \frac{p(p - 3)}{2} \). Here, \( s \) represents the number of yarn segments, and \( p \) is the number of pegs. The task is to determine the number of pegs when there are 27 yarn segments.
To do this, substitute 27 for \( s \) in the equation, result in:

• From the word problem: Convert the real-world context into the formula \( 27 = \frac{p(p - 3)}{2} \).
• Clear the fraction by multiplying the entire equation by 2, leading to a simplified equation: \( 54 = p^2 - 3p \).
• The equation now becomes a quadratic equation once it's rearranged: \( p^2 - 3p - 54 = 0 \).

By recognizing the algebraic representations of the problem, you can follow these steps to reach the solution.

Equations in Mathematics Education

Understanding equations is crucial in mathematics education as they form the basis for expressing relationships between different quantities. Quadratic equations, such as \( ax^2 + bx + c = 0 \), are a specific type of equation frequently encountered in studies and applications.
In our example, the equation \( p^2 - 3p - 54 = 0 \) appears after translating a real-world problem about a wall hanging into mathematical terms. Quadratics are common in numerous real-life applications, making them an essential part of a student's mathematical toolkit.
Equations offer a systematic way to find unknown values, and mastering them initiates a deeper understanding of mathematics:

• Provides a means to represent and solve problems through mathematical language.
• Allows for consistent and reproducible methods to find solutions.
• Forms the foundation for more advanced topics in math, like calculus and linear algebra.

These reasons highlight why equations play such a prominent role in mathematics education.

Quadratic Formula Applications

The quadratic formula is one of the key tools for solving quadratic equations. The formula is: \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides the solutions to the equation \( ax^2 + bx + c = 0 \).
In our exercise, the quadratic equation \( p^2 - 3p - 54 = 0 \) needed solving. By identifying \( a = 1 \), \( b = -3 \), and \( c = -54 \), you can substitute these into the formula. First, calculate the discriminant \( b^2 - 4ac \), which checks if a real solution exists.
Through the calculation:

• Discriminant \(= 225\), which is positive, indicating two real solutions are present.
• Calculating \( p = \frac{-(-3) \pm \sqrt{225}}{2} \) leads to \( p = 9 \) or \( p = -6 \).
• Since only positive integers are valid for pegs, \( p = 9 \) is the solution.

The quadratic formula is invaluable as it ensures solutions are systematically calculated, facilitating applications across tasks involving quadratic equations.