Solutions To The Equation 3(x-4)(x+5) = 0 A Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the solutions to the equation 3(x-4)(x+5) = 0. We will explore the underlying principles, step-by-step methods, and the reasoning behind each step. By the end of this guide, you will have a solid understanding of how to tackle similar equations and interpret the results effectively.

Understanding the Zero Product Property

At the heart of solving this equation lies the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. Mathematically, if a * b = 0, then either a = 0 or b = 0 (or both). This seemingly simple principle is a powerful tool for solving equations where a product is set equal to zero.

In our case, the equation 3(x-4)(x+5) = 0 presents us with a product of three factors: 3, (x-4), and (x+5). According to the Zero Product Property, for the entire product to be zero, at least one of these factors must be zero. Since 3 is a constant and not equal to zero, we focus on the remaining factors: (x-4) and (x+5).

The Zero Product Property is not just a mathematical trick; it is a logical consequence of how multiplication works. If any number is multiplied by zero, the result is always zero. Conversely, if a product is zero, at least one of the numbers being multiplied must be zero. This fundamental concept is crucial in algebra and beyond, providing a reliable method for solving various types of equations.

Step-by-Step Solution

Now, let's apply the Zero Product Property to solve our equation 3(x-4)(x+5) = 0. We've established that either (x-4) = 0 or (x+5) = 0.

  1. Set each factor to zero:

    • x - 4 = 0
    • x + 5 = 0

  2. Solve the first equation, x - 4 = 0: To isolate x, we add 4 to both sides of the equation:

    • x - 4 + 4 = 0 + 4
    • x = 4

  3. Solve the second equation, x + 5 = 0: To isolate x, we subtract 5 from both sides of the equation:

    • x + 5 - 5 = 0 - 5
    • x = -5

Therefore, the solutions to the equation 3(x-4)(x+5) = 0 are x = 4 and x = -5. These are the values of x that make the equation true. We can verify these solutions by substituting them back into the original equation.

The process of setting each factor to zero and solving the resulting equations is a systematic way to find all possible solutions. It is a direct application of the Zero Product Property and ensures that no solution is missed. This method is particularly useful when dealing with equations involving factored expressions.

Verifying the Solutions

It's always a good practice to verify the solutions we obtain to ensure their accuracy. We can do this by substituting each solution back into the original equation and checking if the equation holds true.

  1. Verify x = 4: Substitute x = 4 into the equation 3(x-4)(x+5) = 0:

    • 3(4-4)(4+5) = 0
    • 3(0)(9) = 0
    • 0 = 0 The equation holds true, so x = 4 is a valid solution.

  2. Verify x = -5: Substitute x = -5 into the equation 3(x-4)(x+5) = 0:

    • 3(-5-4)(-5+5) = 0
    • 3(-9)(0) = 0
    • 0 = 0 The equation holds true, so x = -5 is also a valid solution.

By verifying the solutions, we gain confidence in our answer and ensure that we haven't made any errors in our calculations. This step is a crucial part of the problem-solving process and helps to develop a deeper understanding of the equation and its solutions.

Why the Constant Factor Doesn't Matter

You might have noticed that the constant factor 3 in the equation 3(x-4)(x+5) = 0 didn't directly contribute to the solutions. This is because dividing both sides of the equation by 3 gives us (x-4)(x+5) = 0, which has the same solutions. The constant factor only scales the entire expression but doesn't change the values of x that make the product zero.

In general, when solving equations using the Zero Product Property, constant factors can be ignored as they don't affect the roots of the equation. This simplification can make the solving process more efficient, as we can focus solely on the factors that contain the variable.

However, it's important to remember that if the constant factor were to involve the variable x, then it would need to be considered as a potential factor that could equal zero. For example, if the equation were 3x(x-4)(x+5) = 0, then we would also need to consider the case where 3x = 0, which gives us the additional solution x = 0.

Common Mistakes to Avoid

When solving equations using the Zero Product Property, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and solve equations more accurately.

  1. Forgetting the Zero Product Property: The most common mistake is simply not recognizing the applicability of the Zero Product Property. If the equation is in factored form and set equal to zero, this property is your key to finding the solutions.

  2. Incorrectly distributing: Sometimes, students might try to distribute the factors instead of applying the Zero Product Property. While distribution is a valid algebraic technique, it can lead to a more complicated equation that is harder to solve. The Zero Product Property provides a more direct and efficient approach in this case.

  3. Missing solutions: When solving the individual equations, it's crucial to isolate the variable correctly. A mistake in this step can lead to missing one or more solutions. Always double-check your work and ensure that you've considered all possibilities.

  4. Not verifying solutions: As mentioned earlier, verifying the solutions is an essential step. Failing to do so can result in accepting incorrect solutions, especially if there were errors in the solving process.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving equations using the Zero Product Property.

Conclusion

In conclusion, the solutions to the equation 3(x-4)(x+5) = 0 are x = 4 and x = -5. We arrived at these solutions by applying the Zero Product Property, which states that if the product of factors is zero, then at least one of the factors must be zero. We then set each factor involving x to zero and solved the resulting equations.

We also emphasized the importance of verifying the solutions to ensure their accuracy and discussed why constant factors don't affect the solutions in this context. Additionally, we highlighted common mistakes to avoid when using the Zero Product Property.

Understanding and applying the Zero Product Property is a fundamental skill in algebra. By mastering this concept, you will be well-equipped to solve a wide range of equations and tackle more advanced mathematical problems. Remember to practice regularly and apply the principles discussed in this guide to solidify your understanding.