Solving (x-3)(x+9)=-27 A Step-by-Step Guide

Hey everyone! Let's dive into solving the quadratic equation (x-3)(x+9) = -27. This type of problem is super common in algebra, and mastering it can really boost your math skills. We'll break down each step, so you can confidently tackle similar equations in the future. Solving quadratic equations might seem daunting at first, but with a clear method, it becomes much easier. Our mission here is to not only find the solution but also to understand the process thoroughly. Remember, math isn't just about getting the right answer; it's about understanding why that answer is correct. So, grab your pencils, and let's get started on this math adventure!

Understanding the Problem

Before we jump into solving, let’s make sure we fully grasp what the question is asking. The equation (x-3)(x+9) = -27 is a quadratic equation disguised in factored form. A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to this equation are the values of x that make the equation true. These solutions are also known as the roots or zeros of the equation. Our goal is to find the x values that satisfy the given equation. One common mistake is to try and solve the equation directly from its factored form without first expanding and rearranging it into the standard form. To avoid this pitfall, we'll need to expand the product of the binomials on the left side of the equation. This involves using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), to multiply each term in the first binomial by each term in the second binomial. This expansion is a crucial step because it allows us to combine like terms and eventually set the equation equal to zero, which is necessary for solving most quadratic equations. Once we have the equation in standard form, we can explore different methods for finding the solutions, such as factoring, completing the square, or using the quadratic formula. Each method has its strengths and weaknesses, and the best approach may depend on the specific equation we're dealing with.

Step 1: Expanding the Equation

The first thing we need to do is expand the left side of the equation (x-3)(x+9) = -27. We use the distributive property (or the FOIL method) to multiply the two binomials:

(x - 3)(x + 9) = x(x) + x(9) - 3(x) - 3(9)

Let's break this down step by step. First, we multiply the first terms in each binomial: x * x = x². Next, we multiply the outer terms: x * 9 = 9x. Then, we multiply the inner terms: -3 * x = -3x. Finally, we multiply the last terms: -3 * 9 = -27. Combining these terms, we get:

x² + 9x - 3x - 27

Now, we simplify by combining the like terms (the terms with x):

x² + 6x - 27

So, our equation now looks like this:

x² + 6x - 27 = -27

This expansion is a fundamental step in solving quadratic equations. It transforms the equation from a factored form, which is difficult to work with directly, into a polynomial form that we can manipulate more easily. By expanding the equation, we're essentially unraveling the multiplication that was initially performed, allowing us to see all the individual terms that make up the quadratic expression. This step is crucial because it sets the stage for the subsequent steps in the solution process. Without expanding the equation, we wouldn't be able to combine like terms or isolate the variable, which are essential for finding the solutions. It's like trying to assemble a puzzle without first laying out all the pieces; you need to see the individual components before you can put them together. So, mastering the expansion process is key to unlocking the solutions to quadratic equations.

Step 2: Simplifying the Equation

Now that we've expanded the equation, let's simplify it further. We have:

x² + 6x - 27 = -27

To solve a quadratic equation, we need to set it equal to zero. We can do this by adding 27 to both sides of the equation:

x² + 6x - 27 + 27 = -27 + 27

This simplifies to:

x² + 6x = 0

This simplified form is much easier to work with. By setting the equation equal to zero, we've transformed it into a standard form that's amenable to various solution methods. This is a critical step because many techniques for solving quadratic equations, such as factoring and using the quadratic formula, rely on having the equation in this form. Setting the equation to zero allows us to find the values of x that make the expression equal to zero, which are the solutions or roots of the equation. In essence, we're looking for the points where the graph of the quadratic function intersects the x-axis. The act of simplifying the equation is like decluttering a room; we're removing unnecessary terms and consolidating the equation into its most basic form. This makes it easier to see the structure of the equation and identify the next steps needed to solve it. Without this simplification, we might be working with a more complex expression than necessary, which can increase the chances of making errors. So, taking the time to simplify the equation is a wise investment that pays off in the long run.

Step 3: Factoring the Equation

We now have the equation x² + 6x = 0. To solve this, we can factor out the common factor, which is x:

x(x + 6) = 0

Factoring is like reverse-distributing; we're essentially undoing the multiplication process to break the equation down into simpler components. In this case, we've identified that both terms in the equation share a common factor of x, which allows us to rewrite the equation as a product of two factors: x and (x + 6). This is a powerful technique because it transforms the equation from a sum of terms into a product, which makes it much easier to find the solutions. The solutions are the values of x that make either factor equal to zero. Factoring is often the quickest and most straightforward method for solving quadratic equations, especially when the equation can be easily factored. However, not all quadratic equations can be factored neatly, so it's important to be familiar with other solution methods as well. The ability to factor effectively comes with practice, so it's worth spending time honing this skill. It's like learning a new language; the more you practice, the more fluent you become. Factoring is a fundamental skill in algebra, and it's used extensively in many areas of mathematics. So, mastering this technique will not only help you solve quadratic equations but also provide a solid foundation for more advanced mathematical concepts.

Step 4: Finding the Solutions

Now that we have the factored form x(x + 6) = 0, we can use 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 ab = 0, then either a = 0 or b = 0 (or both).

Applying this property to our equation, we have two possibilities:

1. x = 0
2. x + 6 = 0

The first solution is straightforward: x = 0. For the second possibility, we need to solve for x:

x + 6 = 0

Subtract 6 from both sides:

x = -6

So, the solutions to the equation are x = 0 and x = -6. These are the values of x that make the original equation true. When we substitute these values back into the equation, we'll find that both sides of the equation are equal. Finding the solutions is the ultimate goal of solving any equation. It's like reaching the destination after a long journey. In this case, we've used the zero-product property to break down the factored equation into two simpler equations, each of which we can solve independently. This is a powerful technique that allows us to find the solutions efficiently. The zero-product property is a cornerstone of algebra, and it's used extensively in solving various types of equations. It's based on the fundamental principle that zero is a unique number with special properties. Understanding and applying this property correctly is crucial for success in algebra. Now that we've found the solutions, we can confidently say that we've solved the quadratic equation.

Step 5: Checking the Solutions

To make sure we didn't make any mistakes, it's always a good idea to check our solutions. We have two solutions: x = 0 and x = -6. Let's plug them back into the original equation (x - 3)(x + 9) = -27.

Checking x = 0

(0 - 3)(0 + 9) = (-3)(9) = -27

This checks out!

Checking x = -6

(-6 - 3)(-6 + 9) = (-9)(3) = -27

This also checks out!

Both solutions satisfy the original equation. Checking our solutions is like proofreading a document; it's a final step that ensures we haven't made any errors along the way. It's a crucial part of the problem-solving process, and it can save us from submitting incorrect answers. By substituting our solutions back into the original equation, we can verify that they make the equation true. If the equation holds true for a particular value of x, then that value is indeed a solution. If the equation doesn't hold true, then we know we've made a mistake somewhere and need to go back and review our work. Checking solutions is not just about getting the right answer; it's also about building confidence in our work. When we take the time to verify our solutions, we can be sure that we've understood the problem and applied the correct techniques. This can boost our self-assurance and make us more likely to tackle challenging problems in the future. So, always remember to check your solutions – it's a habit that will serve you well in math and beyond.

Conclusion

So, after working through the steps, we found the solutions to the equation (x-3)(x+9) = -27 are x = 0 and x = -6. Therefore, the correct answer from the options provided is C. x = 0. We've shown the step-by-step process of expanding, simplifying, factoring, and applying the zero-product property. Remember, practice makes perfect, so keep solving similar problems to strengthen your skills! You've got this! Solving quadratic equations is a fundamental skill in algebra, and mastering it can open doors to more advanced mathematical concepts. The process we've followed – expanding, simplifying, factoring, and applying the zero-product property – is a versatile approach that can be used to solve a wide range of quadratic equations. By understanding the underlying principles and practicing the techniques, you can become proficient at solving these types of problems. Remember, math is not just about memorizing formulas; it's about understanding the logic and reasoning behind the methods. The more you practice, the more intuitive these methods will become. So, keep challenging yourself with new problems, and don't be afraid to make mistakes – mistakes are opportunities to learn and grow. With perseverance and a solid understanding of the fundamentals, you can conquer any math challenge that comes your way. And remember, the satisfaction of solving a challenging problem is a reward in itself!