Solutions Of The Equation C² = 50 A Comprehensive Guide

Jul 17, 2025 by Jeany 56 views

The equation c² = 50 presents a fundamental concept in mathematics, inviting us to explore the realm of square roots and their implications. In this comprehensive guide, we will delve into the step-by-step process of solving this equation, uncovering the nature of its solutions and their significance within the broader mathematical landscape. Our exploration will not only equip you with the tools to solve this specific equation but also enhance your understanding of quadratic equations and their diverse applications.

Understanding the Basics: Quadratic Equations and Square Roots

To effectively tackle the equation c² = 50, it's crucial to grasp the underlying concepts of quadratic equations and square roots. A quadratic equation is a polynomial equation of the second degree, meaning it contains a term with the variable raised to the power of two (e.g., x²). These equations often have two solutions, representing the points where the corresponding parabola intersects the x-axis.

Square roots, on the other hand, are the inverse operation of squaring a number. The square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 9 is 3 because 3 * 3 = 9. However, it's important to remember that every positive number has two square roots: a positive square root and a negative square root. This is because both 3 * 3 and (-3) * (-3) equal 9.

In the context of the equation c² = 50, we are essentially seeking the values of 'c' that, when squared, result in 50. This involves finding both the positive and negative square roots of 50.

Solving the Equation c² = 50: A Step-by-Step Approach

Now, let's embark on the journey of solving the equation c² = 50. The most direct approach involves utilizing the concept of square roots. Here's a step-by-step breakdown:

Isolate the variable term: In this case, the variable term (c²) is already isolated on the left side of the equation. This simplifies our task, allowing us to proceed directly to the next step.
Take the square root of both sides: To eliminate the square on the variable 'c', we take the square root of both sides of the equation. Remember, we must consider both the positive and negative square roots.

√(c²) = ±√50

This step is crucial as it unveils the two possible solutions for 'c'. The symbol '±' (plus or minus) signifies that we need to account for both the positive and negative roots.
Simplify the square root: The square root of 50 can be simplified by factoring out the largest perfect square that divides 50. In this case, 50 can be factored as 25 * 2, where 25 is a perfect square (5 * 5).

√50 = √(25 * 2) = √25 * √2 = 5√2

This simplification makes the solution more elegant and easier to work with.
Express the solutions: Now we can express the solutions for 'c':

c = ±5√2

This indicates that there are two solutions: c = 5√2 and c = -5√2.

The Two Solutions: Positive and Negative Square Roots

As we've discovered, the equation c² = 50 has two solutions: 5√2 and -5√2. This duality arises from the fundamental property of square roots: both a positive number and its negative counterpart, when squared, yield the same positive result.

c = 5√2: This is the positive square root of 50. When 5√2 is multiplied by itself, it equals 50.
c = -5√2: This is the negative square root of 50. When -5√2 is multiplied by itself, it also equals 50.

These two solutions represent the points where the parabola represented by the equation y = x² - 50 intersects the x-axis. The positive solution corresponds to the point on the right side of the y-axis, while the negative solution corresponds to the point on the left side.

Approximating the Solutions: Decimal Representation

While the solutions 5√2 and -5√2 are the exact solutions in radical form, it's often helpful to approximate them as decimal values for practical applications. Using a calculator, we can approximate the square root of 2 as 1.4142.

Therefore:

c = 5√2 ≈ 5 * 1.4142 ≈ 7.071
c = -5√2 ≈ -5 * 1.4142 ≈ -7.071

These decimal approximations provide a clearer sense of the magnitude of the solutions. We can see that the solutions are approximately 7.071 and -7.071.

Significance and Applications of the Solutions

The solutions to the equation c² = 50, both in radical form (5√2 and -5√2) and decimal form (approximately 7.071 and -7.071), hold significance in various mathematical and real-world contexts.

Geometry: These solutions could represent the side length of a square with an area of 50 square units. The positive solution, 5√2, would be the actual side length, while the negative solution, -5√2, would not have a physical meaning in this context.
Physics: In physics, these solutions could arise in problems involving projectile motion or simple harmonic motion, where the square of a variable is related to a physical quantity. The positive and negative solutions might represent different directions or phases of motion.
Engineering: Engineers often encounter quadratic equations in designing structures, circuits, and other systems. The solutions to these equations can help determine critical parameters and ensure the stability and performance of the system.
Mathematics: More broadly, the process of solving c² = 50 exemplifies a fundamental technique for solving quadratic equations, which are ubiquitous in mathematics and its applications. This understanding lays the foundation for tackling more complex equations and problems.

Beyond the Basics: Exploring Related Concepts

The equation c² = 50 serves as a gateway to exploring related mathematical concepts, further enriching your understanding of algebra and beyond.

Quadratic Formula: For more complex quadratic equations that cannot be easily solved by taking square roots, the quadratic formula provides a general solution. This formula can be used to solve any quadratic equation of the form ax² + bx + c = 0.
Completing the Square: Completing the square is another technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be factored and solved.
Graphing Quadratic Equations: The solutions to a quadratic equation correspond to the x-intercepts of the parabola represented by the equation. Graphing quadratic equations provides a visual representation of the solutions and their relationship to the equation.
Irrational Numbers: The solutions 5√2 and -5√2 are examples of irrational numbers, which cannot be expressed as a simple fraction. Understanding irrational numbers is crucial for a complete grasp of the real number system.

Conclusion: Mastering Quadratic Equations

In this comprehensive guide, we have meticulously explored the solutions of the equation c² = 50, delving into the underlying concepts, step-by-step solution process, and the significance of the solutions in various contexts. By understanding the principles of quadratic equations and square roots, you've not only mastered this specific equation but also gained valuable tools for tackling a wide range of mathematical problems.

Remember, the key to success in mathematics lies in consistent practice and a willingness to explore new concepts. Embrace the challenges, delve deeper into the subject, and you'll unlock the beauty and power of mathematics.

This journey into the realm of quadratic equations has hopefully illuminated the path to mathematical proficiency. Keep exploring, keep learning, and keep solving!