Problem 76 - Algebra and Trigonometry

Question

The sum of the squares of two consecutive even integers is 1252 . Find the integers.

Step-by-Step Solution

Answer

The integers are 24 and 26 or -26 and -24.

Step 1: Define the integers

Let the two consecutive even integers be \( x \) and \( x + 2 \). Since they are even, they differ by 2.

Step 2: Set up a quadratic equation

According to the problem, the sum of their squares is 1252. This can be written as:\[ x^2 + (x + 2)^2 = 1252 \]

Step 3: Expand and simplify the equation

Expand the equation:\[ x^2 + (x^2 + 4x + 4) = 1252 \] Combine like terms:\[ 2x^2 + 4x + 4 = 1252 \]

Step 4: Simplify to form a standard quadratic equation

Subtract 1252 from both sides to set the equation to 0:\[ 2x^2 + 4x + 4 - 1252 = 0 \] Simplify further to:\[ 2x^2 + 4x - 1248 = 0 \] Divide the entire equation by 2:\[ x^2 + 2x - 624 = 0 \]

Step 5: Solve the quadratic equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -624 \):\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-624)}}{2 \cdot 1} \]Calculating further,\[ x = \frac{-2 \pm \sqrt{4 + 2496}}{2} \]\[ x = \frac{-2 \pm \sqrt{2500}}{2} \]\[ x = \frac{-2 \pm 50}{2} \]

Step 6: Calculate possible solutions for x

The possible values for \( x \) are:\[ x = \frac{48}{2} = 24 \] \[ x = \frac{-52}{2} = -26 \]

Step 7: Determine the integers

When \( x = 24 \), the integers are 24 and 26. When \( x = -26 \), the integers are -26 and -24.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra, commonly expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). These equations are called 'quadratic' because they involve terms up to the second power of the unknown variable, \( x \).

To solve quadratic equations, you can use various methods like factoring, completing the square, or applying the quadratic formula. In our exercise, the quadratic equation was set as \( x^2 + 2x - 624 = 0 \). The quadratic formula, \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]helps find solutions by substituting the values of \( a \), \( b \), and \( c \). This formula derives from rearranging the equation into a standard form and solving for zero, giving you the potential solution roots of the equation.

Understanding quadratic equations is not only crucial for algebra but also for various applications in physics and engineering. They represent parabolic curves when graphed, making them integral to modeling real-world scenarios.

Consecutive Integers

Consecutive integers are numbers that follow each other in order, with a specific common difference between them. In the context of the problem, we are dealing with consecutive even integers. Since we focus on even numbers, the difference between any two consecutive even integers is 2.

For example, if you have integers \( x \) and \( x + 2 \), both will satisfy the condition of being consecutive even numbers. It's important to define and express consecutive integers clearly, especially when forming algebraic equations.

• This clarity helps transform word problems into solvable mathematical models.
• Recognizing patterns of numbers frequently leads to simpler computations and insights into the problem's structure.

In our exercise, identifying integer patterns allowed us to set up the initial equation. This formed the basis for further algebraic manipulation, assisting in solving for unknowns precisely.

Squares of Integers

Understanding the concept of "squares of integers" is important in algebraic contexts. The square of an integer is obtained by multiplying the integer by itself. For instance, if \( n \) is an integer, the square is \( n^2 \).

In our exercise dealing with consecutive even integers, we used squares to represent each number: \( x^2 \) and \((x+2)^2\). The problem stated that the sum of these squares is 1252, leading to the equation:\[ x^2 + (x + 2)^2 = 1252 \]When expanded and simplified, it gives the quadratic equation.

• Squares are foundational in expressing geometric areas, especially squares and rectangles.
• They frequently appear in problems dealing with distances and norms in both mathematical and physical contexts.

Grasping how to manipulate squares in equations enhances your ability to solve complex algebraic problems, allowing for deeper insights into mathematical relationships.