The Sum Of A Number And Its Square Is 42

The Sum of a Number and Its Square is 42: Unraveling the Mathematical Mystery

Finding the solution to the equation x² + x = 42 might seem straightforward at first glance. However, delving into this seemingly simple problem opens doors to exploring various mathematical approaches, highlighting the elegance and power of different problem-solving techniques. This article will meticulously guide you through multiple methods to solve this equation, discussing their strengths and weaknesses, and ultimately providing a comprehensive understanding of the underlying mathematical principles.

Understanding the Problem: x² + x = 42

The core of the problem lies in finding a number (represented by 'x') such that when you add the number itself to its square, the result is 42. This translates into the quadratic equation:

x² + x - 42 = 0

This form allows us to leverage powerful tools from algebra to find the solution(s).

Method 1: Factoring the Quadratic Equation

Factoring is a fundamental technique in algebra that involves expressing a quadratic equation as a product of two simpler expressions. In this case, we are looking for two numbers that add up to 1 (the coefficient of 'x') and multiply to -42 (the constant term). These numbers are 7 and -6.

Therefore, we can factor the equation as:

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

This equation is satisfied if either (x + 7) = 0 or (x - 6) = 0. Solving for 'x' in each case gives us two possible solutions:

• x = -7
• x = 6

This method is efficient and provides a clear, concise solution. However, it relies on the ability to easily identify the factors, which might not always be straightforward for more complex quadratic equations.

Method 2: Using the Quadratic Formula

The quadratic formula is a powerful tool that provides a general solution for any quadratic equation of the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

In our equation, x² + x - 42 = 0, we have a = 1, b = 1, and c = -42. Substituting these values into the quadratic formula, we get:

x = [-1 ± √(1² - 4 * 1 * -42)] / (2 * 1) x = [-1 ± √(1 + 168)] / 2 x = [-1 ± √169] / 2 x = [-1 ± 13] / 2

This gives us two solutions:

• x = (-1 + 13) / 2 = 12 / 2 = 6
• x = (-1 - 13) / 2 = -14 / 2 = -7

The quadratic formula is universally applicable, providing a solution even when factoring is difficult or impossible. However, it involves more calculations and can be more prone to errors if not executed carefully.

Method 3: Completing the Square

Completing the square is another algebraic technique used to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Starting with x² + x - 42 = 0, we move the constant term to the right side:

x² + x = 42

To complete the square, we take half of the coefficient of 'x' (which is 1/2), square it (1/4), and add it to both sides:

x² + x + 1/4 = 42 + 1/4 (x + 1/2)² = 169/4

Taking the square root of both sides:

x + 1/2 = ±√(169/4) = ±13/2

Solving for 'x':

• x = -1/2 + 13/2 = 12/2 = 6
• x = -1/2 - 13/2 = -14/2 = -7

Completing the square is a powerful method that offers a systematic approach to solving quadratic equations. It’s particularly useful when dealing with equations that don't factor easily. However, it can be more computationally intensive than factoring.

Method 4: Graphical Representation

A graphical approach provides a visual representation of the solution. By plotting the function y = x² + x - 42, we can identify the x-intercepts, which represent the solutions to the equation x² + x - 42 = 0. The graph will be a parabola, and the x-intercepts will be at x = 6 and x = -7. This method is excellent for visualizing the solutions and understanding the behavior of the quadratic function. However, it relies on accurate graphing and might not be suitable for precise solutions in all cases.

Comparing the Methods: Strengths and Weaknesses

Each method has its own strengths and weaknesses:

Expanding the Understanding: Beyond the Solution

Solving x² + x = 42 provides a foundational understanding of quadratic equations and their solutions. However, the problem can be expanded to explore deeper mathematical concepts:

• Nature of Roots: The discriminant (b² - 4ac) in the quadratic formula determines the nature of the roots. A positive discriminant (as in this case) indicates two distinct real roots. A zero discriminant would indicate one repeated real root, and a negative discriminant would indicate two complex roots.

• Applications of Quadratic Equations: Quadratic equations have wide-ranging applications in various fields, including physics (projectile motion), engineering (structural analysis), and economics (supply and demand curves). Understanding how to solve them is crucial for tackling real-world problems.

• Higher-Order Equations: The principles used to solve quadratic equations can be extended to solve higher-order polynomial equations, although the methods become more complex.

• Numerical Methods: For more complex equations that cannot be solved analytically, numerical methods like Newton-Raphson iteration can be employed to approximate the solutions.

Conclusion: A Deeper Dive into Mathematical Problem Solving

Solving the equation x² + x = 42 reveals more than just two numerical solutions; it unveils a gateway to a deeper understanding of algebra, quadratic functions, and various problem-solving techniques. By mastering these methods, you equip yourself with invaluable tools not only for solving mathematical problems but also for tackling challenges in various other fields. Remember, the key is to choose the most efficient and appropriate method based on the specific problem at hand. The journey of solving this seemingly simple equation underscores the beauty and power of mathematics. Keep exploring, keep questioning, and keep learning!