Finding A Number That Added To Its Square Equals 72 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. We often encounter problems that require us to find unknown numbers based on given conditions. One such intriguing problem involves a number that, when added to its own square, results in 72. This article delves into the process of finding this elusive number, providing a step-by-step explanation and exploring the underlying mathematical concepts.
Understanding the Problem
Before we dive into the solution, let's clearly define the problem. We are looking for a number, let's call it "x", that satisfies the following condition: when this number is added to its own square, the result is 72. Mathematically, we can express this condition as an equation:
x² + x = 72
This equation is a quadratic equation, a type of equation that involves a variable raised to the power of 2. Quadratic equations have a unique structure and can be solved using various methods, which we will explore in the following sections.
Solving the Quadratic Equation
To solve the quadratic equation x² + x = 72, we need to rearrange it into the standard quadratic form, which is:
ax² + bx + c = 0
where a, b, and c are constants. In our case, we can rearrange the equation by subtracting 72 from both sides:
x² + x - 72 = 0
Now, we have a quadratic equation in standard form, where a = 1, b = 1, and c = -72. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, we will use factoring, as it is the most straightforward method for this particular equation.
Factoring the Quadratic Equation
Factoring involves expressing the quadratic equation as a product of two binomials. To factor x² + x - 72 = 0, we need to find two numbers that multiply to -72 and add up to 1 (the coefficient of the x term). These two numbers are 9 and -8, since 9 * -8 = -72 and 9 + (-8) = 1. Therefore, we can factor the equation as:
(x + 9)(x - 8) = 0
For the product of two factors to be zero, at least one of the factors must be zero. This leads to two possible solutions:
- x + 9 = 0 => x = -9
- x - 8 = 0 => x = 8
Thus, the solutions to the quadratic equation are x = -9 and x = 8. These are the numbers that, when added to their own squares, result in 72.
Verifying the Solutions
To ensure that our solutions are correct, we can substitute them back into the original equation and check if they satisfy the condition.
Verifying x = -9
Substituting x = -9 into the original equation, we get:
(-9)² + (-9) = 81 - 9 = 72
This confirms that x = -9 is a valid solution.
Verifying x = 8
Substituting x = 8 into the original equation, we get:
(8)² + 8 = 64 + 8 = 72
This also confirms that x = 8 is a valid solution.
Therefore, the numbers that satisfy the condition are -9 and 8.
Alternative Method: Quadratic Formula
As mentioned earlier, the quadratic formula is another method for solving quadratic equations. The quadratic formula provides a general solution for any quadratic equation in the form ax² + bx + c = 0:
x = (-b ± √(b² - 4ac)) / 2a
In our case, a = 1, b = 1, and c = -72. Substituting these values into the quadratic formula, we get:
x = (-1 ± √(1² - 4 * 1 * -72)) / (2 * 1)
x = (-1 ± √(1 + 288)) / 2
x = (-1 ± √289) / 2
x = (-1 ± 17) / 2
This leads to two solutions:
- x = (-1 + 17) / 2 = 16 / 2 = 8
- x = (-1 - 17) / 2 = -18 / 2 = -9
As we can see, the quadratic formula yields the same solutions as factoring: x = 8 and x = -9.
Conclusion
In this article, we successfully solved the problem of finding a number that, when added to its own square, results in 72. We explored the underlying mathematical concepts, including quadratic equations and the standard quadratic form. We employed the factoring method to find the solutions and verified them by substituting them back into the original equation. We also demonstrated the use of the quadratic formula as an alternative method for solving quadratic equations. The solutions to the problem are -9 and 8, which means that both numbers, when added to their squares, produce the desired result of 72. This problem exemplifies the power of algebraic techniques in solving mathematical puzzles and finding unknown quantities based on given conditions. Understanding and applying these techniques is crucial for success in mathematics and related fields. The ability to translate word problems into mathematical equations and solve them is a valuable skill that can be applied in various real-world scenarios. Whether it's calculating distances, optimizing resources, or modeling complex systems, the principles of algebra and equation solving play a vital role in our understanding of the world around us. In addition to the specific problem we solved, this article highlights the general approach to solving mathematical problems. The key steps include understanding the problem, translating it into a mathematical form, applying appropriate techniques to solve the equation, and verifying the solutions. By following these steps, we can tackle a wide range of mathematical challenges and gain a deeper appreciation for the beauty and power of mathematics. The problem we addressed also serves as a reminder that mathematical problems often have multiple solutions. In this case, we found two numbers that satisfy the given condition. This underscores the importance of considering all possible solutions and verifying them to ensure accuracy. As we continue our mathematical journey, we will encounter more complex problems with even more intricate solutions. However, the fundamental principles we have learned here will continue to guide us and help us unravel the mysteries of the mathematical world.
Real-World Applications of Quadratic Equations
Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields, including physics, engineering, economics, and computer science. Let's explore some examples:
Physics
In physics, quadratic equations are used to describe the motion of objects under constant acceleration, such as projectiles. For example, the height of a ball thrown into the air can be modeled by a quadratic equation that takes into account the initial velocity, the acceleration due to gravity, and the time elapsed. By solving this equation, we can determine the maximum height reached by the ball and the time it takes to hit the ground.
Engineering
Engineers use quadratic equations in various design and construction projects. For instance, when designing bridges and arches, engineers need to calculate the forces and stresses acting on the structure. Quadratic equations can help determine the optimal shape and dimensions of the structure to ensure its stability and safety. Similarly, in electrical engineering, quadratic equations are used to analyze circuits and determine the values of resistors, capacitors, and inductors that will achieve a desired performance.
Economics
In economics, quadratic equations are used to model supply and demand curves. The point where the supply and demand curves intersect represents the equilibrium price and quantity of a product or service. Quadratic equations can also be used to analyze cost functions, revenue functions, and profit functions. By solving these equations, economists can make predictions about market behavior and optimize business decisions.
Computer Science
In computer science, quadratic equations are used in various algorithms and data structures. For example, the time complexity of certain sorting algorithms, such as bubble sort and insertion sort, is quadratic, meaning that the time it takes to sort a list of n items is proportional to n². Quadratic equations are also used in computer graphics to model curves and surfaces. Bezier curves, which are commonly used in computer-aided design (CAD) software and font design, are defined by quadratic equations.
These are just a few examples of the many real-world applications of quadratic equations. The ability to solve quadratic equations is a valuable skill in various professions and industries. By understanding the principles behind quadratic equations and their applications, we can gain a deeper appreciation for the power of mathematics in solving real-world problems.
Conclusion
The problem of finding a number that, when added to its square, results in 72, is a classic example of how quadratic equations can be used to solve mathematical puzzles. By understanding the properties of quadratic equations and applying techniques such as factoring and the quadratic formula, we can successfully find the solutions to these problems. Moreover, we have seen that quadratic equations have numerous real-world applications in various fields, making them an essential tool for scientists, engineers, economists, and computer scientists alike. The ability to solve quadratic equations is a valuable skill that can open doors to a wide range of opportunities and contribute to our understanding of the world around us. As we continue to explore the world of mathematics, we will encounter more complex and challenging problems, but the fundamental principles we have learned here will continue to serve as a foundation for our success. By embracing the power of mathematics, we can unlock new possibilities and make a positive impact on the world.
In summary, the numbers that satisfy the condition are -9 and 8. This article has provided a comprehensive explanation of the solution process, highlighting the importance of understanding quadratic equations and their applications.
