Solving (x-3)(x-5)=35 Find The Solution

Introduction

When faced with the equation (x-3)(x-5)=35, finding the solution involves a step-by-step approach that combines algebraic manipulation and problem-solving techniques. This comprehensive guide will walk you through the process of solving the equation, explaining each step in detail. By understanding the underlying principles, you'll not only be able to solve this particular equation but also similar equations you might encounter in the future. We will explore how to transform the equation into a standard quadratic form, apply factoring or the quadratic formula, and ultimately identify the correct solutions from the given options. The aim is to make the solution process clear, concise, and easily understandable, ensuring that you grasp the fundamental concepts involved.

Understanding the Problem: (x-3)(x-5)=35

To effectively tackle the equation (x-3)(x-5)=35, it's crucial to first understand its structure and the principles involved in solving it. This equation is a quadratic equation presented in a factored form on one side and a constant on the other. The main goal is to find the values of x that satisfy the equation, meaning the values that make the left side equal to the right side. The initial step in solving this type of equation is to expand the factored form to obtain a standard quadratic equation in the form of ax^2 + bx + c = 0. This form is essential because it allows us to apply standard techniques for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Understanding the properties of quadratic equations, such as the relationship between roots and coefficients, is also beneficial in verifying the solutions and gaining a deeper insight into the problem. By breaking down the equation and identifying the necessary steps, we can approach the solution process systematically and accurately.

Step-by-Step Solution

To solve the equation (x-3)(x-5)=35, we'll follow a series of steps to arrive at the solution. The initial step involves expanding the left side of the equation. By applying the distributive property (also known as the FOIL method), we multiply each term in the first parenthesis by each term in the second parenthesis. This gives us: xx - 5x - 3x + 15***, which simplifies to x^2 - 8x + 15. Now our equation looks like this: x^2 - 8x + 15 = 35. The next crucial step is to transform the equation into the standard quadratic form, which is ax^2 + bx + c = 0. To do this, we subtract 35 from both sides of the equation, resulting in x^2 - 8x - 20 = 0. This form is essential because it allows us to use standard methods for solving quadratic equations.

With the equation in standard form, we can now proceed to find the solutions. One common method is factoring, which involves finding two binomials that multiply to give the quadratic expression. In this case, we look for two numbers that multiply to -20 and add to -8. These numbers are -10 and 2. Therefore, we can factor the quadratic expression as (x - 10)(x + 2) = 0. Setting each factor equal to zero gives us two possible solutions: x - 10 = 0 and x + 2 = 0. Solving these equations, we find x = 10 and x = -2. These are the values of x that satisfy the original equation. By systematically expanding, rearranging, and factoring, we've successfully found the solutions to the equation. This step-by-step approach ensures accuracy and clarity in the solution process.

Verifying the Solutions

After finding the possible solutions for the equation (x-3)(x-5)=35, it's important to verify that these solutions are correct. Verification involves substituting each solution back into the original equation to check if it holds true. This step is crucial because it helps to identify any errors made during the solution process and ensures that the values obtained are indeed the correct solutions. Let's start by verifying the first solution, x = 10. Substituting x = 10 into the original equation, we get (10-3)(10-5) = 35. Simplifying this, we have (7)(5) = 35, which is indeed true. This confirms that x = 10 is a valid solution.

Next, we verify the second solution, x = -2. Substituting x = -2 into the original equation, we get (-2-3)(-2-5) = 35. Simplifying this, we have (-5)(-7) = 35, which is also true. This confirms that x = -2 is also a valid solution. By verifying both solutions, we can be confident that our solution process was accurate, and we have correctly identified the values of x that satisfy the equation. This step adds a layer of assurance and completes the problem-solving process effectively. By substituting the values back into the original equation and confirming their validity, we ensure the accuracy of our results.

Analyzing the Answer Choices

Now that we have found the solutions to the equation (x-3)(x-5)=35, which are x = 10 and x = -2, we can analyze the given answer choices to determine the correct one. The answer choices provided are:

A. x = -8 B. x = -5 C. x = 2 D. x = 10

By comparing our solutions with the answer choices, we can clearly see that x = 10 is one of the correct solutions and is present in the answer choices. The other correct solution, x = -2, is not listed among the options. Therefore, the correct answer choice is D. x = 10. Analyzing the answer choices in this way helps to ensure that we select the appropriate solution from the given options. This step is a final check to confirm our work and accurately answer the question.

Conclusion

In conclusion, the process of solving the equation (x-3)(x-5)=35 involves several key steps: expanding the equation, transforming it into standard quadratic form, factoring (or using the quadratic formula), and verifying the solutions. By expanding the original equation, we obtained x^2 - 8x + 15 = 35. Transforming it into the standard form, we got x^2 - 8x - 20 = 0. Factoring this quadratic equation, we found the solutions to be x = 10 and x = -2. We verified these solutions by substituting them back into the original equation, confirming their validity. Upon analyzing the provided answer choices, we identified x = 10 as the correct solution.

Understanding these steps not only helps in solving this particular equation but also provides a solid foundation for tackling other quadratic equations. The systematic approach of expanding, rearranging, factoring, and verifying ensures accuracy and clarity in the problem-solving process. This method is a fundamental skill in algebra and is essential for various mathematical applications. By mastering these techniques, you can confidently solve a wide range of quadratic equations and enhance your mathematical abilities.