Solving P^2 + 12p = 0: A Step-by-Step Guide

Hey guys! Let's dive into solving the quadratic equation p^2 + 12p = 0. Quadratic equations might seem intimidating at first, but trust me, they're super manageable once you understand the basic techniques. This guide will walk you through a step-by-step solution, ensuring you grasp every concept along the way. We'll break down the equation, explore different methods to solve it, and highlight the key principles involved. So, buckle up and let's get started!

Understanding Quadratic Equations

Before we jump into the solution, let's take a moment to understand what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'p') is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. It's super important to recognize this form because it helps us apply various methods to find the solutions or roots of the equation.

In our equation, p^2 + 12p = 0, we can see that 'a' is 1 (the coefficient of p^2), 'b' is 12 (the coefficient of p), and 'c' is 0 (the constant term). Understanding these coefficients is crucial for applying the quadratic formula or factoring techniques. Recognizing that c is zero in this specific case gives us a hint that factoring might be a straightforward method to use. When you're faced with a quadratic equation, always try to identify the coefficients first – it’s like having the key to unlock the solution!

Method 1: Factoring

Factoring is often the quickest and most efficient method for solving quadratic equations, especially when the equation can be easily factored. Our equation, p^2 + 12p = 0, is a prime candidate for factoring. The basic idea behind factoring is to express the quadratic equation as a product of two binomials. This relies on the distributive property in reverse, and it allows us to find the values of 'p' that make the equation true. Factoring transforms a complex-looking equation into a simpler form that is easier to solve.

Step 1: Identify the Common Factor

Look for a common factor in both terms of the equation. In p^2 + 12p = 0, both terms have 'p' as a common factor. This is a crucial observation because pulling out this common factor will drastically simplify the equation. Factoring out the greatest common factor is always the first step in solving equations by factoring. This simplifies the equation, making it easier to work with. Trust me, this simple step can save you a lot of headaches later on!

Step 2: Factor out the Common Factor

Factor out 'p' from both terms: p(p + 12) = 0. This step is where the magic happens. By factoring out 'p', we’ve transformed the quadratic equation into a product of two factors: 'p' and '(p + 12)'. This makes it incredibly straightforward to find the solutions. The equation now states that the product of these two factors is zero, which is a very powerful piece of information.

Step 3: Set Each Factor Equal to Zero

Now, here's the golden rule: If the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero: p = 0 and p + 12 = 0. This principle is the cornerstone of solving equations by factoring. It breaks down the problem into two simpler equations that are easy to solve individually. It’s like splitting a big task into smaller, manageable parts. This is a super common trick in algebra, so make sure you’ve got it down!

Step 4: Solve for p

Solve each equation separately:

• For p = 0, the solution is simply p = 0.
• For p + 12 = 0, subtract 12 from both sides to get p = -12.

And there you have it! We've found two solutions for 'p': 0 and -12. Solving these individual equations is usually the easiest part, especially after you've mastered the factoring steps. These values of 'p' are the roots or solutions of the original quadratic equation. They are the values that, when plugged back into the original equation, will make the equation true.

Method 2: Quadratic Formula (Just for Understanding)

While factoring is the quickest method for this particular equation, it's good to know the quadratic formula as it can solve any quadratic equation, even those that are hard to factor. The quadratic formula is a universal tool in algebra, and it's worth knowing it by heart. It’s your go-to method when factoring seems too difficult or impossible. Think of it as your trusty backup plan!

The quadratic formula is: p = (-b ± √(b^2 - 4ac)) / 2a

Step 1: Identify a, b, and c

In our equation, p^2 + 12p = 0, we have a = 1, b = 12, and c = 0. Identifying these coefficients is the first crucial step. They are the keys that unlock the quadratic formula. Make sure you get the signs right too! A small mistake here can lead to completely wrong answers.

Step 2: Plug the Values into the Formula

Substitute the values into the quadratic formula: p = (-12 ± √(12^2 - 4 * 1 * 0)) / (2 * 1). This step is all about careful substitution. Make sure you replace each variable with its corresponding value correctly. Take your time and double-check your work to avoid any errors. Accuracy here is key to getting the right solutions.

Step 3: Simplify

Simplify the expression:

• p = (-12 ± √(144)) / 2
• p = (-12 ± 12) / 2

Simplifying the expression involves a bit of arithmetic, but it's nothing you can't handle. Start by simplifying the square root and then the rest of the expression. Remember to follow the order of operations (PEMDAS/BODMAS) to avoid any miscalculations. Keep your work neat and organized, and you'll breeze through this step!

Step 4: Find the Two Solutions

Now, we have two cases:

• p = (-12 + 12) / 2 = 0 / 2 = 0
• p = (-12 - 12) / 2 = -24 / 2 = -12

Again, we find the solutions p = 0 and p = -12. The ± sign in the quadratic formula indicates that there are two possible solutions. This is because quadratic equations typically have two roots. Calculating both solutions is straightforward – just split the expression into two cases and solve each one separately. And there you have it – the same solutions we found using factoring!

Conclusion

So, guys, we've successfully solved the quadratic equation p^2 + 12p = 0 using two methods: factoring and the quadratic formula. Factoring was the quicker method in this case, but knowing the quadratic formula gives you a powerful tool for solving any quadratic equation. The solutions are p = 0 and p = -12. Understanding these methods and practicing them will make you a pro at solving quadratic equations. Remember, the key is to understand the underlying principles and apply them systematically. Keep practicing, and you'll become more confident in your algebra skills!

Whether you prefer factoring or the quadratic formula, mastering these techniques will significantly boost your problem-solving skills in algebra. Keep up the great work, and happy solving! Remember, math isn't about memorizing formulas; it's about understanding the concepts and applying them. So, keep practicing and exploring, and you'll find that math can be both challenging and incredibly rewarding.