Solving For X In X² - 12x + 36 = 0 A Step-by-Step Guide

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Hey guys! Today, let's dive into solving a classic quadratic equation. We're going to figure out the value of x in the equation x² - 12x + 36 = 0. This is a pretty common type of problem in algebra, and once you get the hang of it, you'll be solving these like a pro. We will walk through each step to make sure you understand the whole process. Ready? Let's jump in!

Understanding Quadratic Equations

Before we dive headfirst into solving our specific equation, let’s get a quick refresher on what quadratic equations are all about. A quadratic equation is essentially a polynomial equation of the second degree. What does that mean? Well, it means that the highest power of the variable (in our case, x) is 2. The standard form of a quadratic equation looks like this:

ax² + bx + c = 0

Where a, b, and c are constants, and a isn't zero (otherwise, it wouldn't be a quadratic equation, right?). These constants are just numbers. For example, in our equation x² - 12x + 36 = 0, we can see that a = 1, b = -12, and c = 36.

So, why are these equations important? You might be thinking, “Okay, great, another equation… why should I care?” Well, quadratic equations pop up all over the place in real-world applications. Think about physics, where they're used to model projectile motion (like the path of a ball thrown in the air). They're also used in engineering, economics, and computer graphics. Understanding how to solve them opens up a whole world of problem-solving possibilities.

Now, there are a few main ways we can tackle quadratic equations. The most common methods are:

  1. Factoring: This involves breaking down the quadratic expression into two binomials. It's like reverse multiplication. If we can factor the equation, we can easily find the values of x that make the equation true.
  2. Using the Quadratic Formula: This is a surefire method that works for any quadratic equation, even the ones that are tricky to factor. It's a bit more involved, but it always gets the job done.
  3. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. It’s a bit less commonly used than factoring or the quadratic formula, but it’s a valuable technique to know.

For our equation, x² - 12x + 36 = 0, we're going to start with the factoring method because it’s often the quickest way to solve if it works. Factoring involves rewriting the quadratic expression as a product of two binomials. The cool thing about factoring is that it breaks the problem down into smaller, more manageable pieces.

Think of it like this: we're trying to find two expressions that, when multiplied together, give us our original quadratic expression. This method leans heavily on recognizing patterns and understanding how binomials multiply. When you factor, you're essentially reversing the process of expanding brackets (using the distributive property or FOIL method). So, let’s roll up our sleeves and see how we can factor our equation.

Solving by Factoring

The goal here is to rewrite x² - 12x + 36 as a product of two binomials. This means we want to find two expressions that look something like (x + p)(x + q), where p and q are numbers. When we multiply these two binomials together, we should get back our original equation.

So, how do we find these numbers? Here’s the thought process we’ll use:

  1. Look at the constant term: In our equation, the constant term is 36. We need to find two numbers that multiply to give us 36.
  2. Look at the coefficient of the x term: The coefficient of our x term is -12. The same two numbers we found in step 1 must also add up to -12.

Let’s think about the factors of 36. We’ve got:

  • 1 and 36
  • 2 and 18
  • 3 and 12
  • 4 and 9
  • 6 and 6

Now, we need to figure out which pair of these factors can add up to -12. Remember, we need a negative sum, so we should consider negative factors as well.

Looking at our list, we see that -6 and -6 fit the bill perfectly. -6 multiplied by -6 is 36, and -6 plus -6 is -12. Bingo! We’ve found our numbers.

Now we can rewrite our quadratic equation in factored form:

x² - 12x + 36 = (x - 6)(x - 6)

Or, we can write it even more simply as:

x² - 12x + 36 = (x - 6)²

See how we’ve transformed our equation? We’ve taken a quadratic expression and broken it down into a squared binomial. This is a huge step because now we can easily solve for x.

To find the value(s) of x that make the equation true, we set each factor equal to zero. In this case, we only have one unique factor, which is (x - 6). So, we set that equal to zero:

x - 6 = 0

Now, we solve for x by adding 6 to both sides of the equation:

x = 6

And there we have it! The value of x that satisfies the equation x² - 12x + 36 = 0 is x = 6. Because the factor (x - 6) appears twice (as (x - 6)²), we say that x = 6 is a repeated root or a root with multiplicity 2. This just means that the quadratic equation has only one distinct solution.

So, to recap, we've successfully solved our quadratic equation by factoring. We identified the factors of the constant term (36) that also added up to the coefficient of the x term (-12). This allowed us to rewrite the equation in factored form and then easily find the value of x. This method is super efficient when the quadratic equation is factorable, making it a great tool to have in your math arsenal.

Verifying the Solution

Alright, we've found our solution, but it's always a good idea to double-check our work. It's like adding a safety net to ensure we haven't made any sneaky mistakes along the way. So, let's verify that x = 6 actually works in our original equation, x² - 12x + 36 = 0. Verifying our solution will ensure accuracy and boost our confidence in the result. This step is crucial in mathematics, as it confirms that the value we found truly satisfies the equation.

To verify, we simply substitute x = 6 back into the original equation and see if both sides of the equation are equal. This process is straightforward but incredibly important. Here’s how we do it:

Replace x with 6 in the equation x² - 12x + 36 = 0:

(6)² - 12(6) + 36 = 0

Now, let’s break this down step by step:

  1. Calculate (6)²: 6 squared is 6 times 6, which equals 36. So, we have:

    36 - 12(6) + 36 = 0

  2. Next, we handle the multiplication: -12 multiplied by 6 is -72. Our equation now looks like this:

    36 - 72 + 36 = 0

  3. Now, let’s add and subtract the numbers from left to right: 36 minus 72 is -36. So, we have:

    -36 + 36 = 0

  4. Finally, -36 plus 36 equals 0:

    0 = 0

Boom! We’ve got a match. The left side of the equation equals the right side when we substitute x = 6. This confirms that our solution is correct. Verifying the solution is a crucial step in mathematical problem-solving, ensuring that our answer accurately satisfies the original equation. It's like the final piece of the puzzle that gives us complete confidence in our result. By substituting the value back into the equation and checking for equality, we eliminate the possibility of errors and reinforce our understanding of the solution.

By plugging x = 6 into the equation, we’ve shown that it holds true. This gives us peace of mind knowing that we've solved the equation accurately. Remember, it's always worth the extra few minutes to verify your solutions, especially on exams or in important applications. It’s a small step that can save you from making errors and boost your confidence in your problem-solving skills. So, next time you solve an equation, don’t forget to verify your answer!

Alternative Methods: The Quadratic Formula

While factoring is a fantastic method, it’s not always the easiest or most straightforward way to solve quadratic equations. Some equations are just too tricky to factor, and that’s where the quadratic formula comes to the rescue. Think of the quadratic formula as your reliable backup plan – it works for any quadratic equation, no matter how complex. It's a versatile and powerful tool that ensures you can always find the solutions, even when factoring seems impossible. Understanding and mastering the quadratic formula is essential for anyone serious about solving quadratic equations.

The quadratic formula is derived from the method of completing the square, and it gives us a direct way to find the values of x in the standard quadratic equation form:

ax² + bx + c = 0

The formula itself looks like this:

x = [-b ± √(b² - 4ac)] / (2a)

It might look a bit intimidating at first, but trust me, it’s not as scary as it seems. Let’s break it down and see how it works with our equation, x² - 12x + 36 = 0. Using the quadratic formula involves plugging in the coefficients a, b, and c from our equation. This methodical approach ensures we can solve any quadratic equation, regardless of its factorability.

First, we identify our constants:

  • a = 1 (the coefficient of x²)
  • b = -12 (the coefficient of x)
  • c = 36 (the constant term)

Now, we plug these values into the quadratic formula:

x = [-(-12) ± √((-12)² - 4(1)(36))] / (2(1))

Let’s simplify this step by step:

  1. First, we simplify the numerator. The negative of -12 becomes positive 12, so we have:

    x = [12 ± √((-12)² - 4(1)(36))] / (2(1))

  2. Next, we deal with the expression inside the square root. (-12)² is 144, and 4 times 1 times 36 is also 144. So, we get:

    x = [12 ± √(144 - 144)] / (2(1))

  3. Simplifying further, we see that 144 minus 144 is 0:

    x = [12 ± √0] / (2(1))

  4. The square root of 0 is 0, so our equation becomes:

    x = [12 ± 0] / (2(1))

  5. Now, we simplify the denominator: 2 times 1 is 2:

    x = [12 ± 0] / 2

Since adding or subtracting 0 doesn’t change the value, we have:

x = 12 / 2

Finally, we divide 12 by 2 to get:

x = 6

Just like with factoring, we found that x = 6 is the solution. The quadratic formula confirms our previous result, showing the reliability and versatility of this method. This not only validates our earlier solution but also highlights the power of the quadratic formula as a universal problem-solving tool.

So, whether you’re dealing with a tricky quadratic equation that’s hard to factor, or you just want a method that works every time, the quadratic formula is your go-to tool. It might seem a bit complex at first, but with practice, it’ll become second nature. Plus, it’s a fantastic way to double-check your answers if you’ve used factoring or another method. Mastering the quadratic formula equips you with a powerful technique that can tackle any quadratic equation, ensuring you’re well-prepared for any mathematical challenge.

Conclusion

Alright, guys, we've journeyed through solving the quadratic equation x² - 12x + 36 = 0, and we've explored two effective methods: factoring and using the quadratic formula. Factoring allowed us to break down the equation into simpler terms, making it clear that x = 6 is the solution. Then, we reinforced our finding by applying the quadratic formula, which not only confirmed our answer but also showcased its reliability as a universal method for solving quadratic equations.

We started by understanding the anatomy of a quadratic equation and recognizing its standard form, ax² + bx + c = 0. This foundational knowledge helped us identify the coefficients in our equation, setting the stage for both factoring and applying the quadratic formula. Understanding these basics is crucial for mastering quadratic equations and applying them effectively in various contexts.

Factoring proved to be an efficient approach in this case, as we identified that x² - 12x + 36 could be expressed as (x - 6)². Setting the factor (x - 6) equal to zero, we quickly found that x = 6. This method underscores the importance of pattern recognition and algebraic manipulation in problem-solving. Factoring not only provides a solution but also deepens our understanding of the structure of quadratic equations.

To ensure the accuracy of our solution, we verified it by substituting x = 6 back into the original equation. This step is a cornerstone of mathematical practice, providing confidence in our result and highlighting the importance of meticulousness in problem-solving. Verification is more than just a check; it's a reinforcement of the logical steps we've taken and the correctness of our reasoning.

We then turned to the quadratic formula, a powerful tool that guarantees a solution for any quadratic equation. By plugging in the coefficients a, b, and c into the formula, we methodically worked through the calculations to arrive at the same solution: x = 6. This process not only reinforced our understanding of the formula but also demonstrated its versatility and reliability.

Throughout this exploration, we've seen how different methods can lead to the same answer, and how each method offers unique insights into the nature of quadratic equations. Whether you prefer the elegance of factoring or the robustness of the quadratic formula, having both tools in your arsenal equips you to tackle a wide range of problems. These skills extend beyond the classroom, finding applications in various fields that require mathematical problem-solving.

So, the next time you encounter a quadratic equation, remember the steps we've covered. Understand the equation, choose your method wisely, and always verify your solution. With practice and persistence, you'll become a master at solving quadratic equations. Keep up the great work, and happy solving!