Solving The Quadratic Equation 2m² - 3m - 2 = 0 A Comprehensive Guide
In this comprehensive article, we will delve into the process of solving the quadratic equation 2m² - 3m - 2 = 0. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields, including mathematics, physics, engineering, and economics. Understanding how to solve them is essential for anyone seeking a solid foundation in these disciplines. We will explore different methods for finding the solutions (also known as roots) of this equation, including factoring, using the quadratic formula, and completing the square. By the end of this discussion, you will have a clear understanding of how to tackle similar quadratic equations and a deeper appreciation for the underlying mathematical principles.
One of the most straightforward methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. This method is particularly efficient when the quadratic equation has integer roots. To factor the equation 2m² - 3m - 2 = 0, we need to find two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-2), which is -4, and add up to the middle coefficient (-3). These numbers are -4 and 1. We can then rewrite the middle term (-3m) using these numbers:
2m² - 4m + m - 2 = 0
Now, we can factor by grouping:
2m(m - 2) + 1(m - 2) = 0
Notice that (m - 2) is a common factor. Factoring it out, we get:
(2m + 1)(m - 2) = 0
For the product of two factors to be zero, at least one of them must be zero. Therefore, we set each factor equal to zero and solve for m:
2m + 1 = 0 or m - 2 = 0
Solving these linear equations, we find:
m = -1/2 or m = 2
Thus, the solutions to the quadratic equation 2m² - 3m - 2 = 0 are m = -1/2 and m = 2. Factoring is a powerful technique that simplifies the process of finding the roots when applicable, offering a clear and efficient path to the solution.
When factoring is not readily apparent or proves difficult, the quadratic formula provides a reliable method for solving any quadratic equation of the form am² + bm + c = 0. The quadratic formula is given by:
m = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation. For our equation, 2m² - 3m - 2 = 0, we have a = 2, b = -3, and c = -2. Plugging these values into the quadratic formula, we get:
m = (-(-3) ± √((-3)² - 4(2)(-2))) / (2(2))
m = (3 ± √(9 + 16)) / 4
m = (3 ± √25) / 4
m = (3 ± 5) / 4
This gives us two possible solutions:
m = (3 + 5) / 4 = 8 / 4 = 2
m = (3 - 5) / 4 = -2 / 4 = -1/2
As we found using factoring, the solutions are m = 2 and m = -1/2. The quadratic formula is a universally applicable tool, ensuring we can solve any quadratic equation regardless of its complexity. Its power lies in its ability to provide solutions even when factoring is not feasible, making it a cornerstone of quadratic equation solving.
Completing the square is another method for solving quadratic equations. It involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. To complete the square for the equation 2m² - 3m - 2 = 0, we first divide the entire equation by the leading coefficient (2) to make the coefficient of m² equal to 1:
m² - (3/2)m - 1 = 0
Next, we move the constant term to the right side of the equation:
m² - (3/2)m = 1
Now, we need to add a term to both sides of the equation to make the left side a perfect square trinomial. This term is (b/2)², where b is the coefficient of the m term. In this case, b = -3/2, so (b/2)² = (-3/4)² = 9/16. Adding this to both sides, we get:
m² - (3/2)m + 9/16 = 1 + 9/16
Now, the left side is a perfect square:
(m - 3/4)² = 25/16
Taking the square root of both sides, we have:
m - 3/4 = ±√(25/16)
m - 3/4 = ±5/4
Solving for m, we get:
m = 3/4 ± 5/4
This gives us two possible solutions:
m = (3/4 + 5/4) = 8/4 = 2
m = (3/4 - 5/4) = -2/4 = -1/2
Again, we find the solutions m = 2 and m = -1/2. Completing the square is a versatile technique that not only solves quadratic equations but also provides a deeper understanding of the structure of quadratic expressions. It's particularly useful in scenarios where transformations of quadratic forms are required, showcasing its importance beyond simple root-finding.
To ensure the accuracy of our solutions, we can substitute m = 2 and m = -1/2 back into the original equation 2m² - 3m - 2 = 0.
For m = 2:
2(2)² - 3(2) - 2 = 2(4) - 6 - 2 = 8 - 6 - 2 = 0
For m = -1/2:
2(-1/2)² - 3(-1/2) - 2 = 2(1/4) + 3/2 - 2 = 1/2 + 3/2 - 2 = 4/2 - 2 = 2 - 2 = 0
Both solutions satisfy the original equation, confirming their correctness. Verification is a crucial step in the problem-solving process, providing confidence in the accuracy of the results and reinforcing the understanding of the underlying mathematical principles.
Graphically, the solutions to the quadratic equation 2m² - 3m - 2 = 0 represent the x-intercepts (or roots) of the parabola defined by the equation y = 2m² - 3m - 2. The parabola intersects the x-axis at m = -1/2 and m = 2. Graphical interpretation offers a visual understanding of the solutions, connecting algebraic equations to geometric representations and enhancing the overall comprehension of quadratic equations. This connection between algebra and geometry provides a richer perspective on mathematical concepts.
Quadratic equations have numerous applications in real-world scenarios. For example, they are used in physics to model projectile motion, in engineering to design parabolic structures such as bridges and antennas, and in economics to analyze cost and revenue functions. Applications of quadratic equations are widespread, underscoring their practical importance and relevance in various fields. Understanding quadratic equations is essential for solving real-world problems in science, technology, engineering, and mathematics (STEM) disciplines.
In summary, we have successfully solved the quadratic equation 2m² - 3m - 2 = 0 using three different methods: factoring, the quadratic formula, and completing the square. Each method provides a unique approach to finding the solutions, and understanding all three enhances your problem-solving toolkit. The solutions to the equation are m = -1/2 and m = 2. We also verified these solutions and discussed the graphical interpretation and real-world applications of quadratic equations. Conclusion reinforces the key points covered in the article, emphasizing the importance of mastering various methods for solving quadratic equations and appreciating their significance in diverse applications. With a solid grasp of these concepts, you are well-equipped to tackle more complex mathematical challenges and apply these skills in practical contexts.
