Solving Quadratic Equation X² + X - 156 A Step-by-Step Guide

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Hey guys! Let's dive into the fascinating world of quadratic equations and tackle the equation X² + x - 156 = 0 using factorization. This method is super handy for solving these types of equations, and I'm going to break it down step-by-step so it’s crystal clear. Don't worry if it seems a bit daunting at first; we’ll get through it together!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what quadratic equations are all about. At its heart, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. If 'a' were zero, it wouldn't be a quadratic equation anymore, right? It would turn into a linear equation.

In our equation, X² + x - 156 = 0, we can identify the coefficients as follows: a = 1, b = 1, and c = -156. Recognizing these coefficients is the first step in understanding the structure of the equation and how we can manipulate it to find the solutions.

Quadratic equations pop up all over the place in real-world applications, from physics (think projectile motion) to engineering (designing structures) and even finance (modeling growth and decay). So, mastering how to solve them is a seriously valuable skill. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths, but today we're focusing on the factorization method because it's often the quickest and most straightforward when it works. Now, let's get into the nitty-gritty of factorization!

The Factorization Method: A Closer Look

So, what exactly is factorization, and why is it so cool? Factorization is all about breaking down a complex expression into simpler parts—factors—that, when multiplied together, give you the original expression. In the context of quadratic equations, we aim to rewrite the quadratic expression (ax² + bx + c) as a product of two binomials. A binomial, if you recall, is just an algebraic expression with two terms, like (x + p) or (x + q).

The general idea behind factorization is to find two numbers, let's call them p and q, such that their product equals the constant term 'c' and their sum equals the coefficient of the linear term 'b'. In other words, we need to find p and q that satisfy these two conditions:

  1. p * q = c
  2. p + q = b

Once we find these magical numbers, we can rewrite the quadratic equation in the factored form: (x + p)(x + q) = 0. This form is super useful because if the product of two factors is zero, then at least one of the factors must be zero. This is a fundamental principle that allows us to find the solutions (also called roots) of the equation. We simply set each factor equal to zero and solve for 'x'.

Why does this work? It all comes down to the zero-product property, which states that if ab = 0, then either a = 0 or b = 0 (or both). This property is the cornerstone of solving equations by factoring. By factoring the quadratic expression, we transform the problem from finding 'x' that satisfies a quadratic equation to finding 'x' that makes one or both of the binomial factors equal to zero. Trust me; it’s a game-changer once you get the hang of it!

Step-by-Step: Solving X² + x - 156 = 0 by Factorization

Alright, let's get our hands dirty and factorize the equation X² + x - 156 = 0. We'll go through each step meticulously so you can follow along and understand the process. Remember, practice makes perfect, so don't hesitate to try this with other quadratic equations too!

Step 1: Identify the Coefficients

As we discussed earlier, the first step is to identify the coefficients a, b, and c in our equation. In X² + x - 156 = 0, we have:

  • a = 1 (the coefficient of x²)
  • b = 1 (the coefficient of x)
  • c = -156 (the constant term)

Identifying these coefficients correctly is crucial because they guide our search for the numbers p and q in the next step. Make sure you double-check these values before moving on. A small mistake here can throw off the entire solution, so accuracy is key!

Step 2: Find Two Numbers (p and q) that Satisfy the Conditions

This is the heart of the factorization method. We need to find two numbers, p and q, such that:

  • p * q = c = -156
  • p + q = b = 1

Finding these numbers might seem like a puzzle, but there's a systematic way to approach it. Start by listing the factor pairs of 'c' (-156 in our case). Since 'c' is negative, one number in the pair will be positive, and the other will be negative. This is because a positive times a negative gives a negative. Here are some factor pairs of 156 (we'll consider the signs shortly):

  • 1 and 156
  • 2 and 78
  • 3 and 52
  • 4 and 39
  • 6 and 26
  • 12 and 13

Now, we need to consider the signs and see which pair adds up to 'b' (which is 1). We're looking for a pair with a difference of 1 since one number will be positive and the other negative. Looking at the list, the pair 12 and 13 stands out. If we make 12 negative and 13 positive, we have:

  • -12 * 13 = -156
  • -12 + 13 = 1

Bingo! We've found our numbers: p = -12 and q = 13. This step might take some trial and error, but with practice, you'll get quicker at spotting the right pairs.

Step 3: Rewrite the Quadratic Equation in Factored Form

Now that we have our numbers p and q, we can rewrite the quadratic equation in the factored form: (x + p)(x + q) = 0. Plugging in our values, we get:

(x - 12)(x + 13) = 0

This is a crucial step because we've transformed the original quadratic equation into a product of two binomials. Remember, this factored form is equivalent to the original equation, but it's much easier to solve. We're almost there!

Step 4: Apply the Zero-Product Property and Solve for x

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 our case, we have (x - 12)(x + 13) = 0. So, either (x - 12) = 0 or (x + 13) = 0 (or both). Let's solve each of these equations:

  1. x - 12 = 0 Add 12 to both sides: x = 12
  2. x + 13 = 0 Subtract 13 from both sides: x = -13

And there you have it! We've found the solutions to our quadratic equation. The solutions are x = 12 and x = -13.

The Solutions: x = 12 and x = -13

So, after all that hard work, we've successfully solved the quadratic equation X² + x - 156 = 0 using factorization. Our solutions are x = 12 and x = -13. These are the values of 'x' that make the equation true. You can always check your answers by plugging them back into the original equation to make sure they satisfy it.

For example, let's check x = 12:

(12)² + 12 - 156 = 144 + 12 - 156 = 156 - 156 = 0

And for x = -13:

(-13)² + (-13) - 156 = 169 - 13 - 156 = 169 - 169 = 0

Both solutions check out! This gives us confidence that we've done everything correctly.

Understanding the solutions of a quadratic equation is not just about finding numbers; it's about understanding the behavior of the quadratic function. The solutions (also called roots or zeros) represent the points where the parabola (the graph of the quadratic function) intersects the x-axis. In our case, the parabola intersects the x-axis at x = 12 and x = -13.

Tips and Tricks for Factorization

Factorization can sometimes be tricky, especially when dealing with larger numbers or negative signs. But don't worry, I've got a few tips and tricks up my sleeve that can help you become a factorization pro!

  1. Practice Regularly: The more you practice, the better you'll become at spotting factor pairs and recognizing patterns. Try solving a variety of quadratic equations with different coefficients to build your skills.
  2. List Factor Pairs Systematically: When finding the numbers p and q, list the factor pairs of 'c' systematically. This will help you avoid missing any potential pairs and make the process more efficient.
  3. Pay Attention to Signs: Be extra careful with signs, especially when 'c' is negative. Remember that one number in the pair will be positive, and the other will be negative. The sign of 'b' will tell you which number should be positive and which should be negative.
  4. Check Your Solutions: Always check your solutions by plugging them back into the original equation. This will help you catch any mistakes and ensure your answers are correct.
  5. Use the Quadratic Formula When Needed: Factorization is not always the easiest method, especially when the roots are not integers. In such cases, the quadratic formula is your best friend. We'll cover that in another discussion!

Wrapping Up

Guys, we've covered a lot in this guide! We've explored what quadratic equations are, delved into the factorization method, and step-by-step solved the equation X² + x - 156 = 0. Remember, solving quadratic equations by factorization is a powerful tool, and with practice, you'll become more confident and efficient in using it. Keep practicing, and don't hesitate to revisit this guide whenever you need a refresher. Happy solving!

If you have any questions or want to explore more examples, feel free to ask. Math can be challenging, but it's also incredibly rewarding when you crack a tough problem. Keep up the great work!