Solve Quadratic Equations: Factored Form Explained

Hey guys! Today, we're diving deep into the world of quadratic equations, specifically those presented in factored form. Factored form is like a secret code that, once cracked, makes finding the solutions super easy. We'll take a step-by-step approach to not only solve the given equation but also understand the underlying principles. So, grab your thinking caps, and let's get started!

Understanding Factored Form

Before we jump into solving the equation $(x+3)(x-5)=0$, let's break down what factored form actually means. A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' is the variable we want to solve for. Factored form is simply another way of writing a quadratic equation, where the quadratic expression is expressed as a product of two linear factors. In our case, $(x+3)(x-5)$ represents the factored form of a quadratic expression. Think of it as the equation having been pre-simplified for us, making our job a whole lot easier.

So, why is factored form so special? The beauty of factored form lies in a fundamental principle of mathematics known as 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 $A \cdot B = 0$, then either $A = 0$ or $B = 0$ (or both). This seemingly simple property is the key to unlocking the solutions of quadratic equations in factored form. By setting each factor equal to zero, we can quickly find the values of 'x' that make the entire equation true. It's like having a magic key that opens the door to the solutions.

Let's take a closer look at how this works. In our example, $(x+3)(x-5)=0$, we have two factors: $(x+3)$ and $(x-5)$. According to the Zero Product Property, for the product of these factors to be zero, either $(x+3)$ must be zero or $(x-5)$ must be zero (or both). This transforms our single quadratic equation into two simpler linear equations, which are much easier to solve. This is the power of factored form – it simplifies the problem, making it much more manageable. Understanding this concept is crucial for solving not just this equation, but any quadratic equation presented in factored form. It's a fundamental tool in your mathematical toolkit.

Solving the Equation $(x+3)(x-5)=0$

Now that we've grasped the concept of factored form and the Zero Product Property, let's apply this knowledge to solve the equation $(x+3)(x-5)=0$. This is where the magic happens, guys! Remember, the Zero Product Property tells us that if the product of two factors is zero, then at least one of the factors must be zero. So, in our equation, this means either $(x+3) = 0$ or $(x-5) = 0$.

Our next step is to solve each of these linear equations separately. Let's start with the first one: $(x+3) = 0$. To isolate 'x', we need to get rid of the '+3'. We can do this by subtracting 3 from both sides of the equation. This gives us: $x + 3 - 3 = 0 - 3$, which simplifies to $x = -3$. Congratulations! We've found our first solution.

Now, let's move on to the second equation: $(x-5) = 0$. This time, we need to get rid of the '-5'. We can do this by adding 5 to both sides of the equation. This gives us: $x - 5 + 5 = 0 + 5$, which simplifies to $x = 5$. Awesome! We've found our second solution.

So, what does this mean? We've found that the equation $(x+3)(x-5)=0$ has two solutions: $x = -3$ and $x = 5$. These are the values of 'x' that, when plugged back into the original equation, will make the equation true. You can even check this by substituting these values back into the equation and verifying that the result is indeed zero. This process of solving each factor independently is the core of solving quadratic equations in factored form. It's a straightforward and efficient method, making factored form a valuable tool in your problem-solving arsenal. By understanding and applying the Zero Product Property, you can confidently tackle any quadratic equation presented in this form.

Identifying the Correct Solutions

Alright, we've successfully solved the equation $(x+3)(x-5)=0$ and found two solutions: $x = -3$ and $x = 5$. Now, let's match these solutions with the options provided in the question. This is a crucial step to ensure we're selecting the correct answers. Always double-check your work and make sure your solutions align with the given choices.

Looking at the options, we have:

• A. $x=-5$
• B. $x=-3$
• C. $x=0$
• D. $x=3$
• E. $x=5$
• F. The equation has no solution

Comparing our solutions with the options, we can clearly see that:

• $x = -3$ matches option B
• $x = 5$ matches option E

Therefore, the correct solutions are B. $x=-3$ and E. $x=5$. Options A, C, and D are incorrect because they do not match the solutions we derived. Option F is also incorrect because we clearly found two solutions. This step of comparing your solutions with the given options is vital in any math problem. It's a way to confirm your work and avoid careless mistakes. It also reinforces your understanding of the problem and the solution process. So, always take that extra moment to match your answers with the choices provided. It can make all the difference in getting the problem right!

Why the Other Options Are Incorrect

To truly master solving quadratic equations, it's not enough to just find the correct solutions; it's also important to understand why the other options are incorrect. This deeper understanding solidifies your knowledge and helps you avoid common mistakes in the future. Let's analyze why options A, C, D, and F are not solutions to the equation $(x+3)(x-5)=0$.

• A. $x = -5$: If we substitute $x = -5$ into the equation, we get $(-5 + 3)(-5 - 5) = (-2)(-10) = 20$, which is not equal to zero. Therefore, $x = -5$ is not a solution. This highlights the importance of correctly identifying the values that make each factor equal to zero.
• C. $x = 0$: Substituting $x = 0$ into the equation gives us $(0 + 3)(0 - 5) = (3)(-5) = -15$, which is also not equal to zero. This demonstrates that simply guessing a value like zero will not necessarily lead to a solution. The solutions are directly related to the values that make each factor zero.
• D. $x = 3$: Plugging in $x = 3$, we get $(3 + 3)(3 - 5) = (6)(-2) = -12$, which is not zero. This reinforces the idea that the solutions are specific to the factors in the equation and are not arbitrary values.
• F. The equation has no solution: This option is incorrect because we have already found two distinct solutions, $x = -3$ and $x = 5$. Quadratic equations can have zero, one, or two real solutions. In this case, we have two real solutions. Understanding why an equation might have no solution (e.g., when dealing with the square root of a negative number in the quadratic formula) is also important, but this equation clearly has solutions.

By understanding why these options are incorrect, you gain a more complete understanding of the solution process. It's not just about finding the right answer; it's about understanding the underlying principles and avoiding common pitfalls. This critical thinking will serve you well in more complex mathematical problems.

Key Takeaways and Tips for Success

We've covered a lot in this guide, guys! Let's recap the key takeaways and some tips for solving quadratic equations in factored form. This will help you solidify your understanding and tackle similar problems with confidence.

• Factored Form: Recognize that factored form represents a quadratic expression as a product of linear factors. This form is a gift, as it simplifies the solution process.
• Zero Product Property: Remember the Zero Product Property! This is the cornerstone of solving equations in factored form. If $A \cdot B = 0$, then either $A = 0$ or $B = 0$ (or both).
• Solve Each Factor: Set each factor equal to zero and solve the resulting linear equations. This is the core step in finding the solutions.
• Check Your Solutions: Always double-check your solutions by substituting them back into the original equation. This ensures accuracy and helps you catch any errors.
• Understand Incorrect Options: Take the time to understand why the other options are incorrect. This deepens your understanding and prevents future mistakes.

Here are some additional tips for success:

• Practice, Practice, Practice: The more you practice, the more comfortable you'll become with solving quadratic equations in factored form. Work through various examples to build your skills.
• Stay Organized: Keep your work organized and write down each step clearly. This helps prevent errors and makes it easier to review your work.
• Don't Be Afraid to Ask for Help: If you're struggling, don't hesitate to ask your teacher, classmates, or online resources for help. Collaboration can be a powerful learning tool.
• Connect the Concepts: Understand how factored form relates to other forms of quadratic equations, such as standard form. This broader understanding will enhance your problem-solving abilities.

By mastering these key takeaways and tips, you'll be well-equipped to solve quadratic equations in factored form and excel in your math studies. Keep up the great work, and remember, math can be fun!