Solving 7x(x-8)=0 Step-by-Step Solutions And Explanation

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The equation 7x(x-8) = 0 is a quadratic equation presented in a factored form. Solving quadratic equations is a fundamental skill in algebra, with applications spanning various fields, including physics, engineering, and economics. This article provides a comprehensive guide on how to solve this particular equation, delving into the underlying principles and offering step-by-step solutions. We will explore the zero-product property, which is the key to solving equations in this format, and demonstrate how to apply it effectively. Understanding these concepts will equip you with the necessary tools to tackle similar problems and enhance your problem-solving abilities in mathematics. This detailed explanation aims to clarify each step, ensuring that even those new to algebra can grasp the process and confidently solve quadratic equations. By the end of this article, you will not only know the solutions to this specific equation but also understand the broader context and techniques applicable to a wide range of algebraic problems. The ability to solve quadratic equations is a crucial stepping stone in more advanced mathematical concepts, making it an essential skill for anyone pursuing further studies in mathematics, science, or engineering.

To effectively solve the equation 7x(x-8) = 0, it is crucial to first understand the concept of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These roots represent the points where the parabola, which is the graphical representation of a quadratic equation, intersects the x-axis. Quadratic equations can have two, one, or no real solutions, depending on the discriminant (b² - 4ac). When the discriminant is positive, there are two distinct real roots; when it is zero, there is one real root (a repeated root); and when it is negative, there are no real roots (but there are two complex roots). The equation 7x(x-8) = 0 is a special case because it is already factored. Factoring is a common method for solving quadratic equations, as it allows us to utilize the zero-product property, which we will discuss in the next section. Understanding the nature of quadratic equations and their solutions is fundamental for solving a wide range of algebraic problems and is an essential topic in algebra.

The zero-product property is a fundamental principle in algebra that states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is the cornerstone for solving equations that are expressed in factored form, such as our equation 7x(x-8) = 0. Mathematically, the zero-product property can be stated as follows: if AB = 0, then A = 0 or B = 0 (or both). This seemingly simple concept has profound implications for solving algebraic equations. It allows us to break down a complex equation into simpler ones by setting each factor equal to zero and solving for the variable. In the context of quadratic equations, the zero-product property is particularly useful when the equation is factored or can be easily factored. By applying this property, we can transform a single quadratic equation into two linear equations, which are much easier to solve. The zero-product property is not limited to quadratic equations; it can be applied to any equation where the product of factors is equal to zero. Understanding and effectively using this property is essential for solving a wide variety of algebraic problems and is a key skill for anyone studying mathematics. Its simplicity and effectiveness make it a powerful tool in the algebraic toolbox, allowing for efficient solutions to complex equations.

Now, let's apply the zero-product property to solve the equation 7x(x-8) = 0. The equation is already in factored form, which makes it straightforward to apply the property. We have two factors: 7x and (x-8). According to the zero-product property, if the product of these factors is zero, then at least one of them must be zero. Therefore, we set each factor equal to zero and solve for x: First, we set 7x = 0. To solve for x, we divide both sides of the equation by 7: 7x / 7 = 0 / 7, which simplifies to x = 0. So, one solution is x = 0. Next, we set the second factor (x-8) equal to zero: x - 8 = 0. To solve for x, we add 8 to both sides of the equation: x - 8 + 8 = 0 + 8, which simplifies to x = 8. Thus, the second solution is x = 8. Therefore, the solutions to the equation 7x(x-8) = 0 are x = 0 and x = 8. These values of x make the equation true. This step-by-step approach demonstrates how the zero-product property simplifies the process of solving factored quadratic equations, making it a valuable tool in algebra. The clarity of this method ensures that each solution is derived logically and accurately, building confidence in problem-solving abilities.

To provide a clear understanding of the solution process, let's outline the step-by-step solution for the equation 7x(x-8) = 0:

  1. Identify the factors: The equation is already in factored form, with the factors being 7x and (x-8). This simplifies the initial step, as we don't need to factor the equation ourselves.
  2. Apply the zero-product property: According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero.
  3. Set the first factor equal to zero: 7x = 0. This creates our first equation to solve.
  4. Solve for x: To solve 7x = 0, divide both sides by 7: 7x / 7 = 0 / 7. This simplifies to x = 0. So, our first solution is x = 0.
  5. Set the second factor equal to zero: x - 8 = 0. This is our second equation to solve.
  6. Solve for x: To solve x - 8 = 0, add 8 to both sides: x - 8 + 8 = 0 + 8. This simplifies to x = 8. Thus, our second solution is x = 8.
  7. Write the solutions: The solutions to the equation 7x(x-8) = 0 are x = 0 and x = 8. These are the values of x that satisfy the equation.

This step-by-step breakdown provides a methodical approach to solving the equation, making it easier to follow and understand each step. By breaking down the process into manageable steps, we ensure clarity and accuracy in the solution. This method can be applied to similar factored equations, making it a valuable technique for solving algebraic problems.

To ensure the accuracy of our solutions, it is crucial to verify them by substituting them back into the original equation. This process confirms that the values we found for x actually satisfy the equation 7x(x-8) = 0. Let's verify each solution separately:

  1. Verify x = 0: Substitute x = 0 into the original equation: 7(0)(0-8) = 0. Simplifying, we get 7(0)(-8) = 0, which further simplifies to 0 = 0. This is a true statement, so x = 0 is indeed a valid solution.
  2. Verify x = 8: Substitute x = 8 into the original equation: 7(8)(8-8) = 0. Simplifying, we get 7(8)(0) = 0, which further simplifies to 0 = 0. This is also a true statement, confirming that x = 8 is a valid solution.

Both solutions, x = 0 and x = 8, satisfy the original equation. This verification step is an essential practice in mathematics, as it helps to catch any errors that may have occurred during the solving process. By verifying our solutions, we can be confident in the accuracy of our results. This practice not only validates the solutions but also reinforces the understanding of how solutions relate to the original equation. The ability to verify solutions is a valuable skill in mathematics, ensuring the correctness and reliability of our answers.

While the zero-product property is highly effective for solving factored quadratic equations like 7x(x-8) = 0, it is important to be aware of alternative methods for solving quadratic equations in general. These methods include:

  1. Factoring: If a quadratic equation is not already factored, you may be able to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. Once factored, you can apply the zero-product property to find the solutions.
  2. Quadratic Formula: The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is: x = (-b ± √(b² - 4ac)) / 2a. This formula can be used to find the solutions to any quadratic equation, regardless of whether it can be factored or not. It is particularly useful when the equation has irrational or complex solutions.
  3. Completing the Square: Completing the square is a method that involves manipulating the quadratic equation to create a perfect square trinomial. This method can be used to solve any quadratic equation and is also the basis for deriving the quadratic formula. It provides a deeper understanding of the structure of quadratic equations.

Understanding these alternative methods expands your problem-solving toolkit and allows you to choose the most efficient method for a given equation. While the zero-product property is ideal for factored equations, the quadratic formula and completing the square are essential for handling more complex cases. The ability to apply different methods based on the problem's nature is a key skill in mathematics, fostering adaptability and a more comprehensive understanding of quadratic equations. Each method offers a unique perspective on solving quadratic equations, enriching your mathematical foundation.

In conclusion, solving the quadratic equation 7x(x-8) = 0 effectively demonstrates the application of the zero-product property, a fundamental concept in algebra. This property allows us to find the solutions by setting each factor equal to zero, leading to the solutions x = 0 and x = 8. We also emphasized the importance of verifying these solutions by substituting them back into the original equation, ensuring their accuracy. Furthermore, we explored alternative methods for solving quadratic equations, including factoring, the quadratic formula, and completing the square, highlighting the versatility and adaptability required in mathematical problem-solving. Understanding these methods provides a comprehensive approach to tackling various quadratic equations, regardless of their form or complexity. The ability to solve quadratic equations is a crucial skill in mathematics, with applications in numerous fields. By mastering these techniques, you not only gain confidence in your algebraic abilities but also lay a strong foundation for more advanced mathematical concepts. The knowledge and skills acquired through solving equations like 7x(x-8) = 0 are invaluable tools in your mathematical journey, empowering you to approach a wide range of problems with competence and clarity.