Solve $6p^2 - 6p - 36 = 0$ By Factoring: A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the world of quadratic equations and tackling a classic problem: solving by factoring. Specifically, we're going to break down the equation $6p^2 - 6p - 36 = 0$ step-by-step, making sure you not only get the right answer but also understand the why behind each move. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means it has the general form of $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to solve for. The solutions to a quadratic equation are also known as its roots or zeros.

In our case, the equation $6p^2 - 6p - 36 = 0$ perfectly fits this form. Here, 'a' is 6, 'b' is -6, 'c' is -36, and our variable is 'p'. Factoring is just one of the methods we can use to find the values of 'p' that make this equation true. Other methods include using the quadratic formula or completing the square. But for this problem, factoring is the most efficient approach.

Factoring, at its core, is about breaking down a complex expression into simpler components – the factors. Think of it like finding the building blocks of a number. For instance, the factors of 12 are 3 and 4 because 3 multiplied by 4 equals 12. In the same vein, we aim to express the quadratic equation as a product of two binomials. When we set each binomial equal to zero, we can solve for 'p'.

Why Factoring Matters

Now, you might be wondering, why bother with factoring when we have other methods at our disposal? Well, factoring is often the quickest and most straightforward method when it's applicable. It not only gives us the solutions but also provides valuable insights into the structure of the equation. Moreover, mastering factoring is crucial for tackling more advanced mathematical concepts later on.

Factoring helps simplify complex expressions, making them easier to work with. This simplification is essential not only in solving equations but also in various other mathematical applications, such as graphing functions and solving inequalities. It's a fundamental skill that forms the bedrock of algebraic manipulation.

Setting Up for Success: The Importance of Preparation

Before diving headfirst into factoring, it's essential to ensure that the equation is in its standard form, $ax^2 + bx + c = 0$. This standard form allows us to easily identify the coefficients 'a', 'b', and 'c', which are crucial for the factoring process. In our given equation, $6p^2 - 6p - 36 = 0$, we're already in good shape, but there's an extra step we can take to simplify things further.

Step-by-Step Solution

Let's get into the nitty-gritty of solving $6p^2 - 6p - 36 = 0$ by factoring.

1. Simplify the Equation

The first thing I always look for when tackling equations like this is whether there's a common factor we can pull out. And guess what? In $6p^2 - 6p - 36 = 0$, all the terms are divisible by 6! This makes our lives so much easier.

Divide both sides of the equation by 6:

$(6p^2 - 6p - 36) / 6 = 0 / 6$

This simplifies to:

$p^2 - p - 6 = 0$

See how much cleaner that looks? Simplifying the equation first makes the factoring process less cumbersome and reduces the chances of making mistakes. It's like decluttering your workspace before starting a project – it helps you focus and work more efficiently.

2. Factor the Quadratic Expression

Now we have a simpler quadratic expression to factor: $p^2 - p - 6$. We need to find two numbers that:

• Multiply to give us the constant term (-6)
• Add up to give us the coefficient of the 'p' term (-1)

Think of it as a puzzle – we need to find the right pieces that fit together. To do this, let's list the factor pairs of -6: (1, -6), (-1, 6), (2, -3), and (-2, 3). Which of these pairs adds up to -1? Bingo! It's 2 and -3.

So, we can rewrite the quadratic expression as a product of two binomials:

$(p + 2)(p - 3) = 0$

Factoring is like reverse multiplication. We're essentially undoing the distributive property (or the FOIL method, if you're familiar with that). The binomials $(p + 2)$ and $(p - 3)$ are the factors of the quadratic expression $p^2 - p - 6$. When multiplied together, they give us the original expression.

3. Set Each Factor Equal to Zero

The zero-product property is the key here. It states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if $A * B = 0$, then either $A = 0$ or $B = 0$ (or both).

Applying this to our factored equation, $(p + 2)(p - 3) = 0$, we set each factor equal to zero:

$p + 2 = 0$ or $p - 3 = 0$

This step is crucial because it allows us to transform a single quadratic equation into two simpler linear equations, which are much easier to solve.

4. Solve for 'p'

Now we have two simple equations to solve. Let's tackle them one by one.

For $p + 2 = 0$, subtract 2 from both sides:

$p = -2$

For $p - 3 = 0$, add 3 to both sides:

$p = 3$

And there you have it! We've found the two values of 'p' that satisfy the equation. These values are the solutions, roots, or zeros of the quadratic equation.

5. The Solutions

So, the solutions to the equation $6p^2 - 6p - 36 = 0$ are $p = -2$ and $p = 3$. This corresponds to option D. {-2, 3}.

Why This Method Works: The Math Behind It

Let's take a moment to appreciate why factoring works. When we factor a quadratic equation into the form $(p + m)(p + n) = 0$, where 'm' and 'n' are constants, we're essentially finding the values of 'p' that make each factor zero. This is because when either $(p + m)$ or $(p + n)$ is zero, the entire product becomes zero, satisfying the equation.

This method is rooted in the fundamental properties of multiplication and the zero-product property. Understanding the underlying principles not only helps you solve problems but also builds a solid foundation for more advanced mathematical concepts.

Alternative Methods: Exploring Other Options

While factoring is often the quickest method, it's not always applicable. Some quadratic equations are difficult or even impossible to factor using simple techniques. In such cases, we can turn to other methods, such as the quadratic formula or completing the square.

The Quadratic Formula

The quadratic formula is a universal solution for any quadratic equation of the form $ax^2 + bx + c = 0$. It's given by:

$x = (-b ± √(b^2 - 4ac)) / (2a)$

This formula always yields the solutions, regardless of whether the equation is factorable or not. It's a powerful tool in your mathematical arsenal.

Completing the Square

Completing the square is another method that can be used to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be factored easily. This method is particularly useful when dealing with equations that don't factor nicely.

Tips and Tricks for Factoring

Factoring can sometimes feel like a puzzle, but with practice and a few tricks up your sleeve, you'll become a factoring pro. Here are some tips to help you along the way:

• Always look for a greatest common factor (GCF) first. As we saw in our example, simplifying the equation by dividing out the GCF can make the factoring process much easier.
• Practice makes perfect. The more you factor, the better you'll become at recognizing patterns and applying the appropriate techniques.
• Use the 'ac' method. This method involves finding two numbers that multiply to 'ac' and add up to 'b'. It's a systematic approach that can help you factor even complex quadratic expressions.
• Don't be afraid to guess and check. Sometimes, the best way to find the factors is to try different combinations until you find the right one.
• Check your work. After factoring, multiply the factors back together to make sure you get the original expression. This is a simple but effective way to catch errors.

Common Mistakes to Avoid

Factoring can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting to set each factor equal to zero. This is a crucial step in the process, and skipping it will lead to incorrect solutions.
• Making sign errors. Pay close attention to the signs of the coefficients and constants when factoring. A small sign error can throw off the entire solution.
• Factoring incorrectly. Double-check your factors by multiplying them back together to ensure they match the original expression.
• Not simplifying the equation first. As we discussed earlier, simplifying the equation by dividing out the GCF can make factoring much easier.
• Giving up too easily. Factoring can sometimes be challenging, but don't get discouraged. Keep practicing, and you'll get the hang of it.

Real-World Applications of Quadratic Equations

Now, you might be wondering, where do quadratic equations actually show up in the real world? Well, they're everywhere! From physics to engineering to economics, quadratic equations play a vital role in modeling and solving a wide range of problems.

• Physics: Quadratic equations are used to describe the trajectory of projectiles, such as a ball thrown in the air. They also appear in the study of motion under constant acceleration.
• Engineering: Engineers use quadratic equations to design bridges, buildings, and other structures. They also use them in electrical circuit analysis and control systems.
• Economics: Quadratic equations can be used to model supply and demand curves, as well as cost and revenue functions.
• Computer Graphics: Quadratic equations are used to create curves and surfaces in computer graphics and animation.

Conclusion: Mastering Factoring and Beyond

So there you have it! We've successfully solved the equation $6p^2 - 6p - 36 = 0$ by factoring. We walked through each step, from simplifying the equation to setting each factor equal to zero and solving for 'p'. We also discussed why this method works, alternative methods, tips and tricks, common mistakes to avoid, and real-world applications of quadratic equations.

Factoring is a fundamental skill in algebra, and mastering it will not only help you solve equations but also build a strong foundation for more advanced mathematical concepts. Keep practicing, and you'll become a factoring whiz in no time! Remember, math isn't just about finding the right answer; it's about understanding the process and the logic behind it. So, keep exploring, keep questioning, and keep learning!

If you found this guide helpful, give it a thumbs up and share it with your friends. And if you have any questions or topics you'd like me to cover in the future, let me know in the comments below. Happy factoring, guys!