Finding The Equation Sum Of A Number And Its Square Is 42

In the realm of mathematics, we often encounter problems that require us to translate word problems into algebraic equations. These equations then serve as the foundation for solving for unknown values. One such problem involves the relationship between a number and its square. Specifically, we are tasked with finding the equation that represents the statement: "The sum of a number and its square is 42." This seemingly simple statement holds the key to a fascinating mathematical exploration, and by carefully dissecting the statement, we can unlock the appropriate equation. In this article, we will embark on a journey to translate this word problem into a mathematical equation, exploring the underlying concepts and the step-by-step process involved. We will delve into the world of quadratic equations, which often arise when dealing with squares of numbers, and how they can be used to model real-world scenarios. Furthermore, we will analyze the given options and determine which equation accurately captures the relationship described in the problem statement. By the end of this exploration, you will have a clear understanding of how to translate word problems into mathematical expressions and how to identify the correct equation for a given scenario. This skill is not only essential for solving mathematical problems but also for developing critical thinking and problem-solving abilities that can be applied in various aspects of life.

The core of this problem lies in our ability to translate the given statement into a mathematical equation. This involves identifying the key components of the statement and representing them using algebraic symbols. Let's break down the statement piece by piece:

• "A number": This indicates an unknown value, which we can represent using a variable. Let's use the variable x to represent this unknown number.
• "Its square": This refers to the square of the number we just defined, which would be x squared, or x².
• "The sum of": This indicates addition, meaning we need to add the number and its square.
• "Is 42": This signifies equality, telling us that the sum of the number and its square is equal to 42.

Now, let's put these pieces together to form an equation. The sum of the number (x) and its square (x²) is equal to 42. Therefore, the equation that represents this statement is:

x² + x = 42

This equation is a quadratic equation, which is an equation where the highest power of the variable is 2. Quadratic equations are commonly encountered in mathematical problems, especially those involving squares and relationships between numbers. They have a wide range of applications in various fields, including physics, engineering, and economics. Understanding how to form and solve quadratic equations is a fundamental skill in mathematics.

Now that we have derived the equation from the problem statement, let's examine the given options and determine which one matches our result. The options provided are:

• x² + x = 42
• x² + 2x = 42
• x² + x + 42 = 0
• x² + 2x + 42 = 0

Comparing our derived equation (x² + x = 42) with the options, we can clearly see that the first option, x² + x = 42, is the correct equation. This equation accurately represents the statement that the sum of a number and its square is 42.

The other options are incorrect for the following reasons:

• x² + 2x = 42: This equation represents the sum of the square of a number and twice the number, which is not what the problem statement describes.
• x² + x + 42 = 0: This equation is a quadratic equation set equal to zero, but it does not directly translate from the original statement. While this equation can be derived from the correct equation by subtracting 42 from both sides, it is not the most direct representation of the problem statement.
• x² + 2x + 42 = 0: This equation combines the errors of the previous two options, representing the sum of the square of a number, twice the number, and 42, set equal to zero. This equation is not related to the original statement.

Therefore, the only equation that accurately represents the given statement is x² + x = 42.

While the question asks for the equation, let's take it a step further and explore how to solve the equation to find the actual numbers that satisfy the condition. To solve the quadratic equation x² + x = 42, we first need to rearrange it into the standard quadratic form, which is ax² + bx + c = 0. To do this, we subtract 42 from both sides of the equation:

x² + x - 42 = 0

Now, we can solve this quadratic equation using several methods, such as factoring, completing the square, or the quadratic formula. In this case, factoring is the most straightforward method. We need to find two numbers that multiply to -42 and add up to 1 (the coefficient of the x term). These numbers are 7 and -6.

Therefore, we can factor the quadratic equation as follows:

(x + 7)(x - 6) = 0

For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:

x + 7 = 0 or x - 6 = 0

Solving these linear equations, we get:

x = -7 or x = 6

Therefore, the two numbers that satisfy the condition that the sum of the number and its square is 42 are -7 and 6.

Let's verify these solutions:

• For x = -7: (-7)² + (-7) = 49 - 7 = 42
• For x = 6: (6)² + 6 = 36 + 6 = 42

Both solutions satisfy the given condition.

In this exploration, we have successfully translated the statement "The sum of a number and its square is 42" into a mathematical equation. We have seen how to break down a word problem into its constituent parts and represent them using algebraic symbols. We identified the correct equation as x² + x = 42 and also went a step further to solve the equation and find the two numbers that satisfy the condition, which are -7 and 6. This process highlights the importance of understanding the relationship between words and mathematical expressions. Being able to translate word problems into equations is a fundamental skill in mathematics, with applications ranging from solving simple puzzles to modeling complex real-world phenomena. By mastering this skill, you can unlock the power of mathematics to solve problems and gain a deeper understanding of the world around you. The ability to formulate equations from word problems is not just a mathematical skill; it is a valuable tool for critical thinking and problem-solving in various aspects of life. As you continue your mathematical journey, remember that practice and perseverance are key to mastering this art. Keep exploring, keep solving, and keep unlocking the power of mathematics!