Solving $6m^2 - 13m + 6 = 0$ Quadratic Equation A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of solving a quadratic equation. Quadratic equations, a fundamental concept in algebra, are polynomial equations of the second degree. They have the general form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Finding the solutions, also known as roots or zeros, of a quadratic equation is a common task in mathematics and has numerous applications in various fields, including physics, engineering, and economics. We will specifically address the equation 6m2−13m+6=06m^2 - 13m + 6 = 0 and explore different methods to find the value(s) of m that satisfy the equation. This equation represents a classic example of a quadratic equation where the coefficient of the squared term is not 1, which adds an extra layer of complexity to the solving process. Understanding how to solve such equations is crucial for mastering algebraic manipulations and problem-solving skills. Through a step-by-step approach, we will break down the equation, apply suitable techniques, and arrive at the solution. This article aims to provide a clear and concise explanation, making it accessible to both students learning algebra and anyone interested in refreshing their knowledge of quadratic equations.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its own advantages and applicability. The most common methods include factoring, completing the square, and using the quadratic formula. Factoring involves rewriting the quadratic expression as a product of two linear expressions. This method is efficient when the quadratic expression can be easily factored. Completing the square is a more general method that can be used to solve any quadratic equation. It involves manipulating the equation to form a perfect square trinomial. The quadratic formula is a direct formula that provides the solutions of any quadratic equation, regardless of whether it can be factored easily. It is derived from the method of completing the square and is a powerful tool for solving quadratic equations. In the context of our equation, 6m2−13m+6=06m^2 - 13m + 6 = 0, we will primarily focus on the factoring method and the quadratic formula to find the solutions. Each method offers a unique perspective on solving quadratic equations, and understanding their underlying principles is essential for choosing the most appropriate method for a given problem. Moreover, the choice of method can sometimes depend on personal preference or the specific characteristics of the equation. Some equations are more easily factored, while others may require the use of the quadratic formula for a straightforward solution. Therefore, a comprehensive understanding of these methods equips one with the flexibility to tackle a wide range of quadratic equations effectively.

Solving 6m2−13m+6=06m^2 - 13m + 6 = 0 by Factoring

Factoring is a technique used to simplify expressions or solve equations by breaking them down into simpler multiplicative components. Factoring the quadratic equation 6m2−13m+6=06m^2 - 13m + 6 = 0 involves expressing the quadratic expression as a product of two binomials. This method is particularly efficient when the coefficients of the quadratic expression are integers, and the expression can be factored neatly. The first step in factoring this equation is to find two numbers that multiply to the product of the leading coefficient (6) and the constant term (6), which is 36, and add up to the middle coefficient (-13). These two numbers are -4 and -9 because (-4) * (-9) = 36 and (-4) + (-9) = -13. Next, we rewrite the middle term (-13m) using these two numbers: 6m2−4m−9m+6=06m^2 - 4m - 9m + 6 = 0. Now, we factor by grouping. From the first two terms, 6m2−4m6m^2 - 4m, we can factor out 2m2m, resulting in 2m(3m−2)2m(3m - 2). From the last two terms, −9m+6-9m + 6, we can factor out -3, resulting in −3(3m−2)-3(3m - 2). So, the equation becomes 2m(3m−2)−3(3m−2)=02m(3m - 2) - 3(3m - 2) = 0. Notice that (3m−2)(3m - 2) is a common factor, so we can factor it out: (3m−2)(2m−3)=0(3m - 2)(2m - 3) = 0. Finally, we set each factor equal to zero and solve for m: 3m−2=03m - 2 = 0 or 2m−3=02m - 3 = 0. Solving these linear equations gives us the solutions for m. This method of factoring is a powerful tool for solving quadratic equations and provides a clear understanding of the structure of the equation and its solutions.

Detailed Steps for Factoring

Let's break down the factoring process into more detailed steps to ensure clarity. The goal is to rewrite the quadratic expression 6m2−13m+66m^2 - 13m + 6 as a product of two binomials. First, we identify the coefficients: a = 6, b = -13, and c = 6. We need to find two numbers that multiply to a c (which is 6 * 6 = 36) and add up to b (which is -13). These two numbers are -4 and -9. Now, we rewrite the middle term (-13m) using these numbers: 6m2−4m−9m+66m^2 - 4m - 9m + 6. Next, we group the terms into pairs: (6m2−4m)+(−9m+6)(6m^2 - 4m) + (-9m + 6). We factor out the greatest common factor (GCF) from each pair. From the first pair, 6m2−4m6m^2 - 4m, the GCF is 2m2m, so we factor it out: 2m(3m−2)2m(3m - 2). From the second pair, −9m+6-9m + 6, the GCF is -3, so we factor it out: −3(3m−2)-3(3m - 2). Now, we rewrite the equation with the factored pairs: 2m(3m−2)−3(3m−2)=02m(3m - 2) - 3(3m - 2) = 0. We observe that (3m−2)(3m - 2) is a common factor in both terms. We factor out the common binomial factor (3m−2)(3m - 2): (3m−2)(2m−3)=0(3m - 2)(2m - 3) = 0. Finally, we set each factor equal to zero: 3m−2=03m - 2 = 0 and 2m−3=02m - 3 = 0. Solving these linear equations gives us the solutions for m. This step-by-step approach makes the factoring process more manageable and easier to understand. Understanding each step is crucial for applying this method to other quadratic equations.

Finding the Solutions from Factored Form

Once we have factored the quadratic equation into the form (3m−2)(2m−3)=0(3m - 2)(2m - 3) = 0, the next step is to find the values of m that make the equation true. This is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if AB = 0, then either A = 0 or B = 0 (or both). Applying this property to our equation, we set each factor equal to zero: 3m−2=03m - 2 = 0 and 2m−3=02m - 3 = 0. Now, we solve each of these linear equations for m. For the first equation, 3m−2=03m - 2 = 0, we add 2 to both sides: 3m=23m = 2. Then, we divide both sides by 3: m=23m = \frac{2}{3}. For the second equation, 2m−3=02m - 3 = 0, we add 3 to both sides: 2m=32m = 3. Then, we divide both sides by 2: m=32m = \frac{3}{2}. Therefore, the solutions to the quadratic equation 6m2−13m+6=06m^2 - 13m + 6 = 0 are m=23m = \frac{2}{3} and m=32m = \frac{3}{2}. These values of m are the roots of the quadratic equation, and they represent the points where the parabola represented by the quadratic equation intersects the x-axis. Finding these solutions is the primary goal of solving quadratic equations, and the factored form makes this process straightforward. Understanding the zero-product property is key to this step, and it is a fundamental concept in algebra.

Solving 6m2−13m+6=06m^2 - 13m + 6 = 0 using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form ax² + bx + c = 0. The formula is given by: $m = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula provides the solutions for m directly, regardless of whether the quadratic expression can be factored easily. For the equation 6m2−13m+6=06m^2 - 13m + 6 = 0, we identify the coefficients as a = 6, b = -13, and c = 6. We substitute these values into the quadratic formula $m = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(6)(6)}2(6)}$. Now, we simplify the expression step by step. First, we calculate the value inside the square root (−13)2−4(6)(6)=169−144=25(-13)^2 - 4(6)(6) = 169 - 144 = 25. So, the equation becomes: $m = \frac{13 \pm \sqrt{25}12}$. Next, we find the square root of 25, which is 5 $m = \frac{13 \pm 512}$. Now, we have two possible solutions for m $m_1 = \frac{13 + 512} \quad \text{and} \quad m_2 = \frac{13 - 5}{12}$. For m1m_1, we have $m_1 = \frac{1812} = \frac{3}{2}$. For m2m_2, we have $m_2 = \frac{8{12} = \frac{2}{3}$. Therefore, the solutions to the quadratic equation 6m2−13m+6=06m^2 - 13m + 6 = 0 using the quadratic formula are m=32m = \frac{3}{2} and m=23m = \frac{2}{3}, which are the same solutions we obtained by factoring. The quadratic formula is a versatile method that guarantees finding the solutions for any quadratic equation, making it an essential tool in algebra.

Step-by-Step Application of the Quadratic Formula

To ensure a clear understanding of how to use the quadratic formula, let's break down the process into detailed steps. We start with the quadratic equation 6m2−13m+6=06m^2 - 13m + 6 = 0 and the quadratic formula: $m = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$. Step 1 Identify the coefficients. In our equation, a = 6, b = -13, and c = 6. Step 2: Substitute the coefficients into the quadratic formula: $m = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(6)(6)}2(6)}$. Step 3 Simplify the expression inside the square root: (−13)2−4(6)(6)=169−144=25(-13)^2 - 4(6)(6) = 169 - 144 = 25. So, the equation becomes: $m = \frac{13 \pm \sqrt{25}12}$. Step 4 Calculate the square root: $\sqrt{25 = 5$. The equation now is: $m = \frac13 \pm 5}{12}$. Step 5 Separate the equation into two solutions using the plus and minus signs: $m_1 = \frac{13 + 512} \quad \text{and} \quad m_2 = \frac{13 - 5}{12}$. Step 6 Solve for m1m_1: $m_1 = \frac{1812} = \frac{3}{2}$. Step 7 Solve for m2m_2: $m_2 = \frac{8{12} = \frac{2}{3}$. Therefore, the solutions to the quadratic equation are m=32m = \frac{3}{2} and m=23m = \frac{2}{3}. This step-by-step approach ensures that each operation is performed correctly, leading to the accurate solutions. Mastering the quadratic formula involves understanding each of these steps and practicing with various quadratic equations.

Understanding the Discriminant

The discriminant is a crucial part of the quadratic formula, providing valuable information about the nature of the solutions of a quadratic equation. The discriminant is the expression under the square root in the quadratic formula, which is b2−4acb^2 - 4ac. The value of the discriminant determines whether the quadratic equation has two distinct real solutions, one real solution (a repeated root), or two complex solutions. In the equation 6m2−13m+6=06m^2 - 13m + 6 = 0, we have a = 6, b = -13, and c = 6. The discriminant is: $D = b^2 - 4ac = (-13)^2 - 4(6)(6) = 169 - 144 = 25$. Since the discriminant is positive (25 > 0), the quadratic equation has two distinct real solutions. If the discriminant were zero, the equation would have one real solution (a repeated root). If the discriminant were negative, the equation would have two complex solutions. Understanding the discriminant allows us to predict the type of solutions we will obtain before even applying the full quadratic formula. This knowledge can be particularly useful in problem-solving and in understanding the graphical representation of quadratic equations. The discriminant is a fundamental concept in the study of quadratic equations and provides a deeper insight into their properties.

Conclusion

In conclusion, we have successfully solved the quadratic equation 6m2−13m+6=06m^2 - 13m + 6 = 0 using two different methods: factoring and the quadratic formula. Both methods yielded the same solutions: m=23m = \frac{2}{3} and m=32m = \frac{3}{2}. Factoring involved rewriting the quadratic expression as a product of two binomials, setting each factor equal to zero, and solving for m. The quadratic formula provided a direct method for finding the solutions by substituting the coefficients of the quadratic equation into the formula. We also discussed the importance of the discriminant in determining the nature of the solutions. A positive discriminant indicates two distinct real solutions, a zero discriminant indicates one real solution (a repeated root), and a negative discriminant indicates two complex solutions. Mastering these methods and understanding the concepts behind them are essential for solving quadratic equations and for further studies in algebra and related fields. Quadratic equations are a fundamental topic in mathematics with wide-ranging applications, and the ability to solve them is a valuable skill. Whether you choose to factor, use the quadratic formula, or explore other methods, a solid understanding of the underlying principles will enable you to tackle a variety of problems effectively. This article has provided a comprehensive guide to solving 6m2−13m+6=06m^2 - 13m + 6 = 0, and the techniques discussed can be applied to other quadratic equations as well.