Solving Quadratic Equations: X(X-4) = 12

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Solving Quadratic Equations: X(X-4) = 12

Hey guys! Let's dive into solving a quadratic equation today. We're going to break down the equation X(X-4) = 12 step-by-step so that you can not only understand the solution but also apply these techniques to other quadratic equations. Quadratic equations might seem intimidating at first, but with a systematic approach, they become quite manageable. So, grab your calculators, and let’s get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let’s make sure we’re all on the same page about what a quadratic equation actually is. In essence, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually ‘x’) is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

Where ‘a’, ‘b’, and ‘c’ are constants, and ‘a’ is not equal to zero (because if ‘a’ were zero, it would turn into a linear equation). These constants determine the shape and position of the parabola when the equation is graphed.

Why are they important? You might wonder why we even bother with quadratic equations. Well, they pop up in all sorts of real-world scenarios. Think about physics, engineering, economics, and even computer graphics! They help us model projectile motion, design efficient structures, and optimize processes. Understanding how to solve them opens up a world of possibilities.

Different methods for solving: There are several ways to tackle quadratic equations, each with its own strengths and when it's most useful. The main methods include:

  • Factoring: This involves breaking down the quadratic expression into two binomials. It's the quickest method when it works, but it's not always straightforward.
  • Completing the Square: This method transforms the equation into a perfect square trinomial, making it easier to solve. It’s a bit more involved but always works.
  • Quadratic Formula: This is the trusty go-to formula that works for any quadratic equation. It might look a bit scary at first, but it’s a reliable tool in your math arsenal.

So, with these concepts in mind, we're well-equipped to tackle our equation: X(X-4) = 12. Let’s move on to transforming it into the standard form, which is our first crucial step.

Transforming the Equation into Standard Form

Alright, let's get our hands dirty with the equation X(X-4) = 12. The first thing we need to do is transform it into the standard form of a quadratic equation, which, as we discussed, is:

ax² + bx + c = 0

Why do we need to do this? Well, having the equation in standard form makes it much easier to apply the various solving methods we talked about earlier, such as factoring or using the quadratic formula. Think of it as organizing your tools before starting a job – it just makes everything smoother!

Step-by-step transformation: Let’s walk through the steps to get our equation into that nice, neat standard form:

  1. Expand the equation: We start by expanding the left side of the equation. This means distributing the X across the parentheses:

    • X(X-4) becomes X² - 4X

    So, our equation now looks like this:

    • X² - 4X = 12

  2. Move all terms to one side: To get the equation in the standard form, we need to have zero on one side. This means we need to move the 12 from the right side to the left side. We do this by subtracting 12 from both sides:

    • X² - 4X - 12 = 0

    And voila! We now have our equation in the standard form:

    • X² - 4X - 12 = 0

Why is this form important? Now that we have the equation in this form, we can easily identify our coefficients: ‘a’, ‘b’, and ‘c’. In this case:

  • a = 1 (the coefficient of X²)
  • b = -4 (the coefficient of X)
  • c = -12 (the constant term)

Identifying these coefficients is crucial because they are the key ingredients we'll need when we use methods like the quadratic formula. So, with our equation now in standard form, we’re ready to choose our solving method. Let’s consider factoring first, as it's often the quickest route if it works.

Solving by Factoring

Okay, now that our equation is in the standard form (X² - 4X - 12 = 0), let’s see if we can crack it using factoring. Factoring is like reverse multiplication – we're trying to find two binomials that, when multiplied together, give us our quadratic expression. If we can find these binomials, we can easily solve for X.

What is factoring? At its core, factoring involves breaking down a quadratic expression into two simpler expressions (binomials). For example, if we can rewrite X² - 4X - 12 as (X + p)(X + q), where p and q are constants, then we're on the right track. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero.

Step-by-step factoring: Let's factor our equation X² - 4X - 12 = 0:

  1. Find two numbers that multiply to ‘c’ and add up to ‘b’: Remember, in our standard form equation (ax² + bx + c = 0), ‘c’ is -12 and ‘b’ is -4. So, we need to find two numbers that multiply to -12 and add up to -4. Think of the factor pairs of -12: (1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), and (-3, 4). Which pair adds up to -4? You got it – the pair is 2 and -6.

  2. Write the factored form: Now that we have our numbers, we can write the factored form of the equation:

    • (X + 2)(X - 6) = 0

    See how 2 and -6 fit into the binomials? This is the magic of factoring!

  3. Set each factor equal to zero: This is the crucial step where we use the principle that if the product of two factors is zero, then at least one must be zero. So, we set each binomial equal to zero:

    • X + 2 = 0
    • X - 6 = 0

  4. Solve for X: Now we have two simple linear equations to solve:

    • For X + 2 = 0, subtract 2 from both sides:

      • X = -2

    • For X - 6 = 0, add 6 to both sides:

      • X = 6

    So, we have our two solutions: X = -2 and X = 6. Awesome!

When does factoring work best? Factoring is super efficient when the roots of the quadratic equation are integers (whole numbers). It's a quick and elegant method when you can spot the right factors. However, not all quadratic equations can be easily factored. Sometimes, the roots are fractions or irrational numbers, which makes factoring much trickier. In those cases, we might turn to other methods like the quadratic formula.

So, we’ve successfully solved our equation using factoring. But just to be thorough, and to show you another powerful tool, let's also solve it using the quadratic formula. This way, you'll have two methods in your toolkit!

Solving Using the Quadratic Formula

Even though we've already solved X(X-4) = 12 by factoring, let’s tackle it again using the quadratic formula. Why? Because the quadratic formula is like the Swiss Army knife of quadratic equation solving – it works every time, no matter how messy the equation might look. Plus, it’s a great way to double-check our previous answer and reinforce our understanding.

What is the quadratic formula? The quadratic formula is a formula that provides the solutions to any quadratic equation in the standard form ax² + bx + c = 0. The formula looks like this:

X = (-b ± √(b² - 4ac)) / (2a)

Yeah, it looks a bit intimidating at first glance, but trust me, it’s a powerful tool once you get the hang of it. It uses the coefficients a, b, and c from our standard form equation to directly calculate the roots (solutions) of the equation.

Step-by-step using the quadratic formula: Let’s apply this formula to our equation X² - 4X - 12 = 0. Remember, we identified a = 1, b = -4, and c = -12.

  1. Plug in the values: Substitute the values of a, b, and c into the quadratic formula:

    • X = (-(-4) ± √((-4)² - 4 * 1 * -12)) / (2 * 1)

  2. Simplify: Now, let’s simplify the expression step by step:

    • X = (4 ± √(16 + 48)) / 2
    • X = (4 ± √64) / 2
    • X = (4 ± 8) / 2

  3. Calculate the two solutions: The ± sign in the formula means we have two possible solutions, one with addition and one with subtraction:

    • Solution 1 (using +):

      • X = (4 + 8) / 2
      • X = 12 / 2
      • X = 6

    • Solution 2 (using -):

      • X = (4 - 8) / 2
      • X = -4 / 2
      • X = -2

    So, we get the solutions X = 6 and X = -2. Notice anything familiar? These are the exact same solutions we found by factoring! This gives us confidence that we’re on the right track.

When to use the quadratic formula: The quadratic formula is your best friend when factoring seems too difficult or impossible. It works for all quadratic equations, whether the roots are integers, fractions, irrational numbers, or even complex numbers. It’s a bit more computationally intensive than factoring, but its reliability makes it an indispensable tool.

So, we’ve successfully used the quadratic formula to solve our equation, confirming our solutions from the factoring method. Now that we have our solutions, let's take a moment to interpret what they mean in the context of the original equation.

Interpreting the Solutions

We've done the math, and we've found that the solutions to the equation X(X-4) = 12 are X = 6 and X = -2. But what do these solutions actually mean? It's not just about getting the right numbers; it's about understanding what those numbers represent in the context of the equation.

What do the solutions represent? In the context of a quadratic equation, the solutions (also called roots or zeros) are the values of X that make the equation true. In graphical terms, if we were to plot the graph of the equation Y = X² - 4X - 12, the solutions X = 6 and X = -2 would be the points where the parabola intersects the X-axis. These are the X-values where Y equals zero.

Verifying the solutions: One of the best ways to ensure we’ve solved the equation correctly is to plug our solutions back into the original equation and see if they hold true. Let’s do that:

  1. For X = 6:

    • Original equation: X(X-4) = 12
    • Substitute X = 6: 6(6-4) = 12
    • Simplify: 6(2) = 12
    • Result: 12 = 12 (This is true!)

  2. For X = -2:

    • Original equation: X(X-4) = 12
    • Substitute X = -2: -2(-2-4) = 12
    • Simplify: -2(-6) = 12
    • Result: 12 = 12 (This is also true!)

Since both solutions make the original equation true, we can confidently say that we've found the correct values.

Practical implications: While this equation might seem purely mathematical, understanding the solutions can have practical implications in real-world scenarios. For example, if this equation represented a physical system (like the trajectory of a projectile), the solutions could tell us at what points the projectile reaches a certain height or position. In engineering, these solutions might help determine the dimensions needed for a structure to meet specific requirements.

So, by interpreting the solutions, we're not just finding numbers; we're gaining insights into the underlying relationships and behaviors described by the equation. With that understanding, let's wrap up with a quick recap of what we've learned and some key takeaways.

Conclusion and Key Takeaways

Alright, guys, we’ve reached the end of our journey through solving the quadratic equation X(X-4) = 12! We’ve covered a lot of ground, from understanding what quadratic equations are to applying different methods to find their solutions. Let’s quickly recap what we’ve learned and highlight some key takeaways:

  • Understanding Quadratic Equations: We started by defining what a quadratic equation is (a polynomial equation of the second degree) and why they’re important in various fields like physics, engineering, and economics.
  • Transforming to Standard Form: We transformed our equation into the standard form (ax² + bx + c = 0), which is crucial for applying solving methods. For our equation, X² - 4X - 12 = 0, we identified a = 1, b = -4, and c = -12.
  • Solving by Factoring: We successfully factored the equation into (X + 2)(X - 6) = 0 and found the solutions X = -2 and X = 6. Factoring is efficient when the roots are integers.
  • Solving Using the Quadratic Formula: We then used the quadratic formula, X = (-b ± √(b² - 4ac)) / (2a), to solve the same equation and confirmed our solutions as X = -2 and X = 6. The quadratic formula is reliable for all quadratic equations.
  • Interpreting the Solutions: We verified that our solutions made the original equation true and discussed how these solutions represent the points where the parabola intersects the X-axis. We also touched on the practical implications of these solutions in real-world contexts.

Key Takeaways:

  • Multiple Methods: There are multiple methods to solve quadratic equations, each with its own strengths. Factoring is quick when applicable, while the quadratic formula is universally reliable.
  • Standard Form is Key: Transforming the equation into standard form (ax² + bx + c = 0) is essential for both factoring and using the quadratic formula.
  • Solutions Represent Meaning: The solutions to a quadratic equation represent specific values that make the equation true and can have practical interpretations in real-world scenarios.

So, there you have it! We've not only solved the equation X(X-4) = 12, but we've also deepened our understanding of quadratic equations and the methods to solve them. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Keep up the great work, guys!