Solving Quadratic Equations By Factoring The Equation 6x² + X = 7

Jul 11, 2025 by Admin 66 views

Solving Quadratic Equations by Factoring: A Step-by-Step Guide to 6x² + x = 7

Introduction

In the realm of algebra, quadratic equations hold a prominent position. They appear in various scientific and engineering applications, making mastering the techniques to solve them crucial. Among the different methods available, factoring stands out as an elegant and efficient approach for certain quadratic equations. In this comprehensive guide, we will delve into the process of solving the quadratic equation 6x² + x = 7 by factoring, providing a step-by-step explanation to enhance your understanding and problem-solving skills. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. They generally take the form ax² + bx + c = 0, where a, b, and c are constants. Solving quadratic equations involves finding the values of the variable (usually denoted as x) that satisfy the equation, i.e., the values that make the equation true. These values are also known as the roots or solutions of the equation. Factoring is a technique used to decompose a quadratic expression into the product of two linear expressions. This method is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero. By factoring a quadratic equation, we can set each factor equal to zero and solve for the variable, thereby finding the roots of the equation. Factoring is a powerful technique for solving quadratic equations, but it is not always applicable. It works best when the quadratic expression can be factored easily. For equations that are difficult to factor, other methods such as the quadratic formula or completing the square may be more suitable. However, when factoring is possible, it is often the quickest and most straightforward method. The equation 6x² + x = 7 is a quadratic equation that can be solved by factoring. In the following sections, we will walk through the steps of factoring this equation to find its solutions.

1. Transforming the Equation: Setting it to Zero

Before we embark on the factoring process, it is crucial to rearrange the given equation, 6x² + x = 7, into the standard quadratic form, which is ax² + bx + c = 0. This form is essential for applying factoring techniques effectively. To achieve this, we need to move all terms to one side of the equation, leaving zero on the other side. This is done by subtracting 7 from both sides of the equation. The original equation is 6x² + x = 7. To set the equation to zero, subtract 7 from both sides: 6x² + x - 7 = 7 - 7. This simplifies to 6x² + x - 7 = 0. Now the equation is in the standard quadratic form, with a = 6, b = 1, and c = -7. This form allows us to clearly identify the coefficients and constant term, which are necessary for the next step of factoring. Setting the equation to zero is a fundamental step in solving quadratic equations by factoring. It allows us to utilize 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. By setting the equation to zero, we can factor the quadratic expression and then set each factor equal to zero to find the solutions. This step ensures that we are working with an equation that is conducive to factoring and that we can apply the zero-product property correctly. The rearranged equation, 6x² + x - 7 = 0, is now ready for factoring. In the next section, we will explore the process of factoring the quadratic expression 6x² + x - 7 into two linear factors. This will involve finding two binomials that, when multiplied together, give us the original quadratic expression. Factoring is a key step in solving quadratic equations, and mastering this technique is essential for success in algebra and beyond. The ability to transform an equation into its standard form is a foundational skill in algebra. It not only prepares the equation for factoring but also makes it easier to apply other solution methods, such as the quadratic formula or completing the square. This step highlights the importance of algebraic manipulation in problem-solving and sets the stage for the subsequent steps in solving the quadratic equation.

2. Factoring the Quadratic Expression: Unveiling the Binomial Factors

Now that we have the equation in the standard form, 6x² + x - 7 = 0, the next step is to factor the quadratic expression 6x² + x - 7. Factoring involves decomposing the quadratic expression into the product of two linear expressions (binomials). This process requires finding two binomials that, when multiplied together, give us the original quadratic expression. To factor 6x² + x - 7, we need to find two numbers that multiply to give the product of the leading coefficient (6) and the constant term (-7), which is -42, and add up to the coefficient of the middle term (1). These two numbers are 7 and -6, since 7 * -6 = -42 and 7 + (-6) = 1. Now, we rewrite the middle term (x) as the sum of the terms 7x and -6x: 6x² + 7x - 6x - 7 = 0. Next, we factor by grouping. We group the first two terms and the last two terms: (6x² + 7x) + (-6x - 7) = 0. We factor out the greatest common factor (GCF) from each group: x(6x + 7) - 1(6x + 7) = 0. Notice that both terms now have a common factor of (6x + 7). We factor out this common factor: (6x + 7)(x - 1) = 0. We have now successfully factored the quadratic expression 6x² + x - 7 into the product of two binomials: (6x + 7) and (x - 1). This factoring process is a critical step in solving the quadratic equation because it allows us to apply the zero-product property, which we will discuss in the next section. Factoring quadratic expressions is a fundamental skill in algebra. It involves finding the right combination of factors that, when multiplied, give the original expression. This process may require some trial and error, but with practice, it becomes more intuitive. The ability to factor quadratic expressions is essential for solving quadratic equations and for simplifying algebraic expressions in general. The factoring process we used here, factoring by grouping, is a common technique for factoring quadratic expressions with a leading coefficient other than 1. It involves finding two numbers that satisfy the conditions mentioned above, rewriting the middle term using these numbers, and then factoring by grouping. This method is a powerful tool for factoring quadratic expressions and is widely used in algebra. The factored form of the equation, (6x + 7)(x - 1) = 0, is now ready for the final step of solving the quadratic equation: applying the zero-product property to find the values of x that make the equation true.

3. Applying the Zero-Product Property: Finding the Solutions

With the quadratic equation factored into the form (6x + 7)(x - 1) = 0, we can now apply the zero-product property. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if A * B = 0, then either A = 0 or B = 0 (or both). Applying this property to our factored equation, we set each factor equal to zero and solve for x. First, we set the factor (6x + 7) equal to zero: 6x + 7 = 0. To solve for x, we subtract 7 from both sides: 6x = -7. Then, we divide both sides by 6: x = -7/6. This is one solution to the quadratic equation. Next, we set the factor (x - 1) equal to zero: x - 1 = 0. To solve for x, we add 1 to both sides: x = 1. This is the second solution to the quadratic equation. Therefore, the solutions to the quadratic equation 6x² + x = 7 are x = -7/6 and x = 1. These are the values of x that make the equation true. The zero-product property is a fundamental principle in algebra that allows us to solve equations that are factored. It is based on the fact that zero is a unique number in that any number multiplied by zero is zero. This property is particularly useful for solving quadratic equations, as it allows us to break down a quadratic equation into two linear equations, which are much easier to solve. The solutions we found, x = -7/6 and x = 1, are the roots of the quadratic equation. They are the points where the graph of the quadratic function y = 6x² + x - 7 intersects the x-axis. These solutions can be verified by substituting them back into the original equation and checking that the equation holds true. For example, substituting x = -7/6 into the equation 6x² + x = 7, we get: 6(-7/6)² + (-7/6) = 6(49/36) - 7/6 = 49/6 - 7/6 = 42/6 = 7. This confirms that x = -7/6 is a solution. Similarly, substituting x = 1 into the equation, we get: 6(1)² + 1 = 6 + 1 = 7. This confirms that x = 1 is also a solution. The zero-product property is a powerful tool for solving factored equations, and it is essential for solving quadratic equations by factoring. By setting each factor equal to zero and solving for the variable, we can find all the solutions to the equation. This property, combined with the ability to factor quadratic expressions, provides a complete method for solving quadratic equations by factoring.

Conclusion: Mastering Quadratic Equations Through Factoring

In this comprehensive guide, we have meticulously explored the process of solving the quadratic equation 6x² + x = 7 by factoring. We began by transforming the equation into its standard form, 6x² + x - 7 = 0, which is a crucial step for applying factoring techniques. Then, we delved into the factoring process, breaking down the quadratic expression 6x² + x - 7 into its constituent binomial factors, (6x + 7) and (x - 1). This involved finding two numbers that multiply to -42 and add up to 1, rewriting the middle term, and factoring by grouping. Finally, we applied the zero-product property, setting each factor equal to zero and solving for x, to arrive at the solutions x = -7/6 and x = 1. These solutions are the roots of the quadratic equation, representing the values of x that satisfy the equation. Factoring is a powerful and elegant method for solving quadratic equations, particularly when the quadratic expression can be factored easily. It relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. By factoring the quadratic expression, we can transform the quadratic equation into two linear equations, which are much simpler to solve. While factoring is not always the most efficient method for solving all quadratic equations, it is a valuable tool in the algebraist's arsenal. Other methods, such as the quadratic formula and completing the square, are available for equations that are difficult to factor. However, when factoring is possible, it often provides the quickest and most straightforward solution. Mastering the technique of solving quadratic equations by factoring requires practice and a solid understanding of algebraic principles. It involves not only the mechanics of factoring but also the ability to recognize when factoring is an appropriate method and to apply the zero-product property correctly. By working through examples like 6x² + x = 7, you can develop your skills and confidence in solving quadratic equations by factoring. The ability to solve quadratic equations is a fundamental skill in mathematics and has applications in various fields, including physics, engineering, and economics. Quadratic equations model many real-world phenomena, and being able to solve them allows us to understand and predict these phenomena. Factoring is just one of the techniques available for solving quadratic equations, but it is a valuable one to master. By understanding the principles behind factoring and practicing the steps involved, you can become proficient in solving quadratic equations and apply this skill to a wide range of problems.