Problem 24 - Elementary Algebra

Question

For two consecutive positive even integers, the product of the smaller and twice the larger is \(160 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 8 and 10.

Step 1: Define the integers

Let the smaller even integer be \( x \). Since the integers are consecutive even numbers, the larger integer will be \( x+2 \).

Step 2: Set up the equation

According to the problem, the product of the smaller integer and twice the larger integer is 160. This leads to the equation: \( x \cdot 2(x + 2) = 160 \).

Step 3: Simplify the equation

Expand the equation: \( x \cdot (2x + 4) = 160 \). Simplifying this, we get \( 2x^2 + 4x = 160 \).

Step 4: Rearrange into a standard quadratic form

Bring everything to one side of the equation to get: \( 2x^2 + 4x - 160 = 0 \). This is a standard quadratic equation.

Step 5: Simplify the quadratic equation

Divide the entire equation by 2 to make it simpler: \( x^2 + 2x - 80 = 0 \).

Step 6: Solve the quadratic equation using the quadratic formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -80 \). Plug into the formula: \( x = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times (-80)}}{2 \times 1} \).

Step 7: Calculate the discriminant

Calculate the discriminant: \( 2^2 - 4 \times 1 \times (-80) = 4 + 320 = 324 \).

Step 8: Find the square root of the discriminant

Compute \( \sqrt{324} = 18 \). Now substitute back into the formula.

Step 9: Solve for x

Substitute into the quadratic formula: \( x = \frac{-2 \pm 18}{2} \). This yields two values, \( x = \frac{16}{2} = 8 \) and \( x = \frac{-20}{2} = -10 \). Since we need positive integers, choose \( x = 8 \).

Step 10: Find the two integers

With \( x = 8 \), the integers are 8 and \( 8 + 2 = 10 \).

Key Concepts

Consecutive Even Integers

Understanding consecutive even integers is crucial when working with certain algebra problems, especially when they involve finding unknown values.
Even integers are numbers like -4, 0, 6, or 14. They are divisible by 2 with no remainder. Consecutive even integers simply mean even numbers that follow one after the other. For example, 2 and 4, or 10 and 12.
To express this in algebraic terms, if we take an even integer as \( x \), the next consecutive even integer is \( x+2 \). This is because integers are spaced by a difference of 2.
In the given problem, identifying these two even numbers helps create an equation based on their relationships, such as their product, sum, or difference. Remember, the key is setting up the right expressions based on the problem's wording.

Quadratic Formula

The quadratic formula is a vital tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). In this formula, \( a \), \( b \), and \( c \) are coefficients of the equation.
The quadratic formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula helps find the values of \( x \) by determining the roots of the quadratic equation.
When using this formula, it's crucial first to ensure the equation is set in the standard form \( ax^2 + bx + c = 0 \), which is done by arranging like terms and ensuring all terms are on one side of the equation.
In practice, you calculate the discriminant \( \Delta = b^2 - 4ac \). The value of the discriminant provides information about the nature and number of the roots, such as whether they are real or complex.
Using the quadratic formula was a central part of solving our exercise. Once the quadratic equation was derived, the formula provided an efficient way to find the integer solutions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill for solving various math problems, including coefficients and variables.
It involves reorganizing and condensing expressions to their simplest form, often making complex equations easier to handle. In algebra, this might mean:

• Expanding brackets, such as using the distributive property to transition from \( x(2x + 4) \) to \( 2x^2 + 4x \).
• Combining like terms, such as adding or subtracting coefficients that share the same variable form.

The key is to methodically follow these steps, maintaining a logical approach to ensure every equation or expression transition leads to simpler or more manageable terms.
In the example, simplifying was pivotal in transitioning from the original product equation to a standard quadratic form before applying further problem-solving strategies.

Solving Word Problems in Algebra

Solving word problems in algebra can initially seem challenging, but it becomes much more manageable once broken into clear steps.
The first task is to comprehend the problem, identifying all given information and what the problem is asking. This guides you in setting up the right equations.

• Identify the variables: Define what the unknowns are, often using algebraic symbols like \( x \).
• Translate words to math: Convert the sentences into algebraic equations. This is where understanding terms like 'product', 'sum', and others is crucial.
• Solve the equation: Use appropriate mathematical techniques, such as the quadratic formula or simplifying algebraic expressions, to find the solution.

In our exercise, recognizing it involved consecutive even integers allowed us to set up expressions reflecting the problem's conditions, ultimately leading to solving the quadratic equation.