Solving For M Two Possible Values In The Equation (5 M^2-25)/2=50

In this article, we will delve into the process of solving quadratic equations, specifically focusing on finding the possible values of a variable. We will use a concrete example to illustrate the steps involved in isolating the variable and arriving at the solutions. Quadratic equations, characterized by the presence of a squared term, often have two solutions, which represent the points where the quadratic function intersects the x-axis. Understanding how to solve these equations is fundamental in various fields, including mathematics, physics, engineering, and economics. This article aims to provide a clear and comprehensive guide to solving quadratic equations, empowering readers to confidently tackle similar problems. We'll break down each step in detail, ensuring a solid grasp of the underlying concepts. Let's embark on this journey to unravel the solutions of quadratic equations!

Before diving into the solution, let's first understand quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions can be found using various methods, including factoring, completing the square, and the quadratic formula. Understanding the properties of quadratic equations is crucial for solving them effectively. For instance, the discriminant, given by b² - 4ac, provides insights into the nature of the roots: a positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. Recognizing the structure of a quadratic equation and the relationships between its coefficients and roots is essential for choosing the appropriate solution method. In this article, we will focus on solving a specific quadratic equation by isolating the variable and applying algebraic techniques. By mastering these techniques, you'll gain a solid foundation for tackling more complex quadratic equation problems.

Our goal is to determine the two possible values of m that satisfy the equation:

$$\frac{5m^2 - 25}{2} = 50$$

This equation is a quadratic equation in disguise. To reveal its quadratic nature, we need to manipulate the equation algebraically. The presence of the m² term signals that we are dealing with a quadratic equation, and we anticipate finding two solutions for m. The challenge lies in isolating m and simplifying the equation to a standard form that allows us to easily identify the solutions. We'll employ a step-by-step approach, carefully applying algebraic operations to both sides of the equation to maintain equality. Each step will bring us closer to isolating m and uncovering its possible values. By meticulously following the steps, we'll not only find the solutions but also gain a deeper understanding of the algebraic techniques involved in solving quadratic equations. Let's begin by clearing the fraction and simplifying the equation to set the stage for finding the values of m.

Let's embark on the journey of finding the two possible values of m for the given equation:

$$\frac{5m^2 - 25}{2} = 50$$

Step 1: Clear the Fraction

To eliminate the fraction, we multiply both sides of the equation by 2:

$$2 * \frac{5m^2 - 25}{2} = 2 * 50$$

This simplifies to:

$$5m^2 - 25 = 100$$

Clearing the fraction is a crucial first step as it simplifies the equation and makes it easier to manipulate. By multiplying both sides by the denominator, we effectively remove the fraction, allowing us to work with whole numbers and simplify the subsequent steps. This step is a common technique in solving equations involving fractions and is essential for isolating the variable. Now that we've cleared the fraction, we can proceed to isolate the term containing m² and further simplify the equation.

Step 2: Isolate the Term with m²

Next, we add 25 to both sides of the equation to isolate the term containing m²:

$$5m^2 - 25 + 25 = 100 + 25$$

This gives us:

$$5m^2 = 125$$

Isolating the term with m² is a fundamental step in solving for m. By adding 25 to both sides, we effectively move the constant term to the right side of the equation, leaving the term with m² isolated on the left side. This step prepares the equation for the next operation, which involves dividing both sides by the coefficient of m². Isolating the variable term is a common strategy in solving various types of equations and is crucial for ultimately finding the value(s) of the variable.

Step 3: Divide by the Coefficient of m²

Now, we divide both sides of the equation by 5:

$$\frac{5m^2}{5} = \frac{125}{5}$$

This simplifies to:

$$m^2 = 25$$

Dividing both sides by the coefficient of m² isolates m² on one side of the equation. This step is crucial for undoing the multiplication and bringing us closer to finding the value(s) of m. By dividing by 5, we effectively reduce the equation to a simpler form where m² is equal to a constant. This simplified form allows us to easily take the square root of both sides, which is the next step in solving for m. Dividing by the coefficient is a common technique in solving algebraic equations and is essential for isolating the variable.

Step 4: Take the Square Root

To find the values of m, we take the square root of both sides of the equation:

$$\sqrt{m^2} = \pm \sqrt{25}$$

This gives us two possible values for m:

$$m = \pm 5$$

Taking the square root of both sides is the final step in solving for m. It's crucial to remember that taking the square root yields both positive and negative solutions. In this case, the square root of 25 is both 5 and -5, as both values, when squared, result in 25. This highlights the fact that quadratic equations often have two solutions. Recognizing and including both positive and negative roots is essential for accurately solving quadratic equations. We have now successfully found the two possible values of m that satisfy the given equation.

Therefore, the two possible values of m are:

• m = 5
• m = -5

These two values, 5 and -5, are the solutions to the quadratic equation. They represent the points where the equation holds true. We have successfully found these solutions by systematically applying algebraic techniques, including clearing the fraction, isolating the variable term, dividing by the coefficient, and taking the square root. Each step played a crucial role in simplifying the equation and revealing the possible values of m. Understanding these steps and the underlying principles is essential for solving various quadratic equations. The ability to confidently find the solutions to quadratic equations is a valuable skill in mathematics and its applications.

To ensure our solutions are correct, let's substitute each value of m back into the original equation:

For m = 5:

$$\frac{5(5)^2 - 25}{2} = \frac{5(25) - 25}{2} = \frac{125 - 25}{2} = \frac{100}{2} = 50$$

For m = -5:

$$\frac{5(-5)^2 - 25}{2} = \frac{5(25) - 25}{2} = \frac{125 - 25}{2} = \frac{100}{2} = 50$$

Both values satisfy the original equation, confirming our solutions are correct. Verification is a crucial step in problem-solving as it ensures the accuracy of the results. By substituting the solutions back into the original equation, we can check if they hold true. This process helps to identify any potential errors and reinforces our understanding of the solution. In this case, both m = 5 and m = -5 satisfy the equation, providing confidence in our solution.

In this article, we successfully determined the two possible values of m that satisfy the equation $rac{5m^2 - 25}{2} = 50$. By systematically applying algebraic techniques, we isolated m and found the solutions to be m = 5 and m = -5. We also verified these solutions by substituting them back into the original equation. This exercise demonstrates the importance of understanding quadratic equations and the methods for solving them. Quadratic equations are fundamental in various mathematical and scientific contexts, and mastering their solution techniques is essential for success in these fields. The step-by-step approach we followed in this article provides a clear and concise guide for solving similar problems. By understanding each step and the underlying principles, you can confidently tackle a wide range of quadratic equations. Keep practicing and applying these techniques to further enhance your problem-solving skills.