Solve $5m^2 + 3m - 10 = 0$ Quadratic Equation

Dec 7, 2025 by Alex Johnson 46 views

Solving the Quadratic Equation $5m^2 + 3m - 10 = 0$

Quadratic equations are a fundamental part of algebra, and solving quadratic equations like $5m^2 + 3m - 10 = 0$ is a skill that opens doors to understanding many mathematical concepts and real-world applications. When you encounter an equation in the form $am^2 + bm + c = 0$ , where 'a', 'b', and 'c' are coefficients and 'm' is the variable you're trying to find, you're dealing with a quadratic equation. The beauty of these equations lies in their predictable structure, which allows for systematic methods of finding their solutions, also known as roots. These roots represent the values of 'm' that make the equation true. For the specific equation $5m^2 + 3m - 10 = 0$ , we have $a=5$ , $b=3$ , and $c=-10$ . There are several ways to tackle this, but the most universally applicable method is the quadratic formula. This formula is a powerful tool derived from completing the square and is guaranteed to give you the solutions for any quadratic equation. Understanding how to apply it correctly, and simplifying the results, is key to mastering this area of mathematics.

The Power of the Quadratic Formula

The quadratic formula is your best friend when it comes to solving quadratic equations that might not be easily factorable. It provides a direct route to the solutions, no matter how complex they might seem. The formula itself is: $m = rac{-b ext{ ± } ext{√}(b^2 - 4ac)}{2a}$ . This formula might look a bit intimidating at first glance, but let's break it down. The term inside the square root, $b^2 - 4ac$ , is called the discriminant. The discriminant tells us about the nature of the roots: if it's positive, you'll have two distinct real roots; if it's zero, you'll have exactly one real root (a repeated root); and if it's negative, you'll have two complex conjugate roots. For our equation, $5m^2 + 3m - 10 = 0$ , we substitute the values of $a=5$ , $b=3$ , and $c=-10$ into the quadratic formula. So, we get $m = rac{-3 ext{ ± } ext{√}(3^2 - 4 imes 5 imes -10)}{2 imes 5}$ . This step is crucial: carefully plug in your coefficients, paying close attention to the signs. A common mistake is misplacing a negative sign, which can drastically alter the outcome. Once you've substituted, the next phase is simplification, which we'll delve into next. The elegance of the quadratic formula is that it encapsulates the entire process of solving a quadratic, eliminating the need to constantly check if factoring is possible or to go through the lengthy process of completing the square each time. It's a robust and reliable method for finding all solutions to the equation.

Simplifying the Solutions

After applying the quadratic formula, the next critical step is to simplify your answer. This involves performing the arithmetic operations within the formula and reducing any radicals to their simplest form. Let's continue with our equation, $5m^2 + 3m - 10 = 0$ . We had: $m = rac{-3 ext{ ± } ext{√}(3^2 - 4 imes 5 imes -10)}{2 imes 5}$ . First, let's calculate the discriminant: $3^2 - 4 imes 5 imes -10 = 9 - (-200) = 9 + 200 = 209$ . So, the formula becomes: $m = rac{-3 ext{ \pm } ext{\sqrt}(209)}{10}$ . Now, we need to check if the square root of 209 can be simplified. To do this, we look for perfect square factors of 209. We can test small prime numbers: 209 is not divisible by 2, 3 (sum of digits is 11), or 5. Let's try 7: $209 ext{ ÷ } 7 ext{ ≈ } 29.8$ . Try 11: $209 ext{ ÷ } 11 = 19$ . Since 11 and 19 are both prime numbers, and neither is a perfect square, the square root of 209 cannot be simplified further. Therefore, the two solutions for 'm' are: $m_1 = rac{-3 + ext{√}(209)}{10}$ and $m_2 = rac{-3 - ext{√}(209)}{10}$ . These are the exact, simplified solutions to the equation $5m^2 + 3m - 10 = 0$ . It's important to present answers in their simplest form, as requested, to demonstrate a thorough understanding of algebraic manipulation and number properties. This process of simplification ensures that the final answer is as concise and understandable as possible, making it easier to use in further calculations or analyses. Simplifying the answer is not just about aesthetics; it's about mathematical rigor.

Understanding the Roots

When we talk about finding all solutions to the equation, we're referring to the roots, which are the values of the variable that satisfy the equation. For a quadratic equation like $5m^2 + 3m - 10 = 0$ , there can be up to two distinct real roots, one repeated real root, or two complex conjugate roots. The nature of these roots is determined by the discriminant ( $b^2 - 4ac$ ). In our case, the discriminant was 209, which is a positive number. This tells us that our equation has two distinct real roots. These roots, $m_1 = rac{-3 + ext{√}(209)}{10}$ and $m_2 = rac{-3 - ext{√}(209)}{10}$ , are the points where the parabola represented by the function $y = 5m^2 + 3m - 10$ would intersect the x-axis if we were to graph it. Understanding these roots is crucial in various fields. In physics, they might represent times when an object reaches a certain height or position. In engineering, they could signify critical values in a system's stability. Even in economics, quadratic equations appear in models for cost, revenue, and profit. The ability to accurately find all solutions and interpret them provides valuable insights into the behavior of systems modeled by quadratic relationships. It’s a foundational skill that allows us to move beyond abstract numbers and apply mathematical principles to tangible problems. The process of deriving these roots ensures that we have a complete picture of the equation's behavior and its potential applications.

Alternative Methods: Factoring and Completing the Square

While the quadratic formula is a reliable method for solving quadratic equations, it's worth noting that other techniques exist, primarily factoring and completing the square. Factoring is often the quickest method, but it only works when the quadratic expression can be easily broken down into two binomials. For $5m^2 + 3m - 10 = 0$ , we could try to find two numbers that multiply to $5 imes -10 = -50$ and add up to $3$ . After some trial and error, you'd find that such integer pairs don't exist, indicating that factoring with integers won't work here. Completing the square is another method that, like the quadratic formula, works for all quadratic equations. It involves manipulating the equation to create a perfect square trinomial. For $5m^2 + 3m - 10 = 0$ , you'd first divide by 5 to get $m^2 + rac{3}{5}m - 2 = 0$ . Then, move the constant term: $m^2 + rac{3}{5}m = 2$ . To complete the square, take half of the coefficient of 'm' (which is $rac{3}{10}$ ), square it ( $rac{9}{100}$ ), and add it to both sides: $m^2 + rac{3}{5}m + rac{9}{100} = 2 + rac{9}{100}$ . This simplifies to $(m + rac{3}{10})^2 = rac{209}{100}$ . Taking the square root of both sides gives $m + rac{3}{10} = ext{\pm} rac{ ext{\sqrt}(209)}{10}$ , and finally, $m = - rac{3}{10} ext{ \pm } rac{ ext{\sqrt}(209)}{10}$ , which combines to $m = rac{-3 ext{ \pm } ext{\sqrt}(209)}{10}$ . While completing the square is a valuable exercise for understanding the derivation of the quadratic formula, using the formula directly is often more efficient for finding all solutions. Each method offers a different perspective on solving quadratic equations, enriching your mathematical toolkit.

Conclusion

Mastering the art of solving quadratic equations is a vital step in mathematical proficiency. For the equation $5m^2 + 3m - 10 = 0$ , we successfully employed the quadratic formula to find all solutions, arriving at $m = rac{-3 ext{ \pm } ext{\sqrt}(209)}{10}$ . The process involved careful substitution, calculation of the discriminant, and rigorous simplification of the answer. Understanding the nature of the roots, as dictated by the discriminant, provides deeper insights into the equation's behavior and its potential applications in various real-world scenarios. While alternative methods like factoring and completing the square exist, the quadratic formula remains the most robust and universally applicable tool for tackling any quadratic equation. Continuous practice with these methods will undoubtedly enhance your problem-solving skills and confidence in algebra. For further exploration into quadratic equations and their properties, you can visit Khan Academy's comprehensive resources on quadratic equations. Additionally, resources like Brilliant.org's math courses offer interactive learning experiences to deepen your understanding.