Problem 161 - Elementary Algebra

Question

Make up a problem involving the product of two consecutive odd integers. Start by choosing two consecutive odd integers. (a) What are your integers? (b) What is the product of your integers? (c) Solve the equation \(n(n+2)=p,\) where \(p\) is the product you found in part (b). (1) Did you get the numbers you started with?

Step-by-Step Solution

Answer

The consecutive odd integers are 7 and 9. Their product is 63.

Step 1: Choose Two Consecutive Odd Integers

Select two consecutive odd integers. For example, let the integers be 7 and 9.

Step 2: Recall the Odd Integers

For part (a), the chosen consecutive odd integers are 7 and 9.

Step 3: Find the Product of the Chosen Integers

Calculate the product of 7 and 9: \[7 \times 9 = 63\].

Step 4: Set Up the Equation

Use the equation provided where the product \(p\) is equal to 63. Substitute into the equation: \[ n(n + 2) = 63 \].

Step 5: Solve the Quadratic Equation

Solve for \(n\) using the quadratic equation: \[ n^2 + 2n - 63 = 0 \]. Use the quadratic formula \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 2\), and \(c = -63\).

Step 6: Calculate the Discriminant

Calculate the discriminant: \[ \Delta = b^2 - 4ac = 2^2 - 4(1)(-63) = 4 + 252 = 256 \].

Step 7: Find the Roots

Determine the roots: \[ n = \frac{-2 \pm \sqrt{256}}{2(1)} \Rightarrow n = \frac{-2 \pm 16}{2} \Rightarrow n = 7 \text{ or } n = -9 \].

Step 8: Verify the Numbers

The roots found are 7 and -9. Since the consecutive odd integers are positive, choose 7. The integers are 7 and 9.

Key Concepts

quadratic equations

To solve the original problem involving the product of two consecutive odd integers, we employ quadratic equations. A quadratic equation is any equation that can be written in the standard form: \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). Quadratic equations have different methods of solution, such as factoring, completing the square, and using the quadratic formula. The quadratic formula is particularly useful and is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where the symbol \( \pm \) means that there will be two solutions, one for each sign.
Let's break down how we used the quadratic formula in our problem:

• We started with the equation \( n(n + 2) = 63 \).
• First, we expressed this in the standard quadratic form: \( n^2 + 2n - 63 = 0 \).
• Here, \( a = 1 \), \( b = 2 \), and \( c = -63 \).
• We found the discriminant \( \Delta = b^2 - 4ac \) to determine if real solutions exist. In this case, \( \Delta = 256 \), which is positive, indicating two real solutions.
• Using the quadratic formula, we calculated the roots as \( n = 7 \) and \( n = -9 \).

Understanding the steps of using a quadratic equation will greatly help tackle similar problems.

product of integers

In our problem, we focused on the product of two consecutive odd integers. Consecutive odd integers follow each other with a sequence difference of 2. For example, if one odd integer is \( n \), the next consecutive odd integer would be \( n + 2 \).

Let's explore the key steps around products of integers:

• Selecting the consecutive odd integers, such as 7 and 9.
• Calculating their product: \( 7 \times 9 = 63 \).
• Substituting the product into the quadratic equation: \( n(n + 2) = 63 \).

Remember, when dealing with consecutive odd integers, their sequence ensures that they can be expressed in a simplified form, making it easier to set up and solve the resulting product equations.

solving equations

Solving equations is a fundamental algebraic skill. In this particular exercise, we solved a quadratic equation derived from the product of integers. Here’s a structured approach to solving such equations:

• Identify the type of equation you have. In our case, we identified the quadratic equation.
• Rearrange the equation into the standard form: \( ax^2 + bx + c = 0 \).
• Calculate the discriminant \( \Delta = b^2 - 4ac \) to check the nature of the roots. A positive discriminant indicates two real and distinct solutions.
• Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions.
• Interpret the roots accordingly. Since our problem involved finding integers, we chose the integer solutions - in this case, 7.

Solving equations often requires systematic steps and checking your results to ensure the solutions align with the physical context of the problem (like positive integers in our case of consecutive odd integers). With practice, solving such equations becomes intuitive and straightforward.