Solving $x^2 - 6x + 12 = 0$: A Step-by-Step Guide

by Andrew McMorgan 50 views

Hey guys! Today, we're diving into the world of quadratic equations to tackle the problem: x2−6x+12=0x^2 - 6x + 12 = 0. This is a classic example that might seem tricky at first glance, but don't worry, we'll break it down step-by-step so you can ace it. We'll explore why some of the given answer choices are incorrect and pinpoint the correct solution. So, let's get started and make math a little less daunting, shall we?

Understanding the Quadratic Equation

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form is ax2+bx+c=0ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. In our case, we have x2−6x+12=0x^2 - 6x + 12 = 0, where a = 1, b = -6, and c = 12. To solve quadratic equations, we're essentially finding the values of x that satisfy the equation. These values are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, but for this problem, the quadratic formula is our best bet. This method ensures we capture both real and complex solutions, which is crucial since our equation might not have straightforward real roots. When approaching such problems, always remember to double-check the coefficients and apply the formula meticulously to avoid common mistakes. This careful approach is especially important in exams where time and accuracy are of the essence. Understanding the nature of the discriminant will also provide insights into the types of roots to expect – whether they are real and distinct, real and equal, or complex conjugates. So, keep your focus sharp and let's get into the nitty-gritty of the quadratic formula!

The Quadratic Formula: Our Key Tool

The quadratic formula is a powerful tool that provides a direct method for finding the solutions to any quadratic equation. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions for x are given by:

x = rac{-b extbf{±} ext{√}(b^2 - 4ac)}{2a}

This formula is a lifesaver when factoring isn't straightforward, and it works for all quadratic equations, regardless of whether the solutions are real or complex. Now, let's identify why this formula is so important. First, it's universally applicable – meaning it works for any quadratic equation you throw at it. No need to guess and check with factoring or complete the square (though those methods are valuable too!). Second, it elegantly handles cases with complex solutions, which are common but can be easily missed if you're only looking for real numbers. So, mastering the quadratic formula is like having a Swiss Army knife in your math toolkit. Remember, the formula itself is a condensed version of completing the square, so understanding its derivation can give you a deeper appreciation for its power. To really make it stick, practice applying it to various equations – you’ll soon see how it simplifies even the trickiest problems. Let's use it now for our equation.

Applying the Quadratic Formula to x2−6x+12=0x^2 - 6x + 12 = 0

Let's apply the quadratic formula to our equation, x2−6x+12=0x^2 - 6x + 12 = 0. We've already identified that a = 1, b = -6, and c = 12. Plugging these values into the formula, we get:

x = rac{-(-6) extbf{±} ext{√}((-6)^2 - 4(1)(12))}{2(1)}

Now, let's simplify step by step. First, we have −(−6)-(-6) which simplifies to 6. Then, inside the square root, we have (−6)2(-6)^2 which equals 36, and 4(1)(12)4(1)(12) which equals 48. So, our equation now looks like this:

x = rac{6 extbf{±} ext{√}(36 - 48)}{2}

Next, we simplify the expression inside the square root: 36 - 48 = -12. This gives us:

x = rac{6 extbf{±} ext{√}(-12)}{2}

Notice that we have a negative number inside the square root, which means we're dealing with complex solutions. This is a key step, as it tells us the solutions will involve the imaginary unit i. Don't let this intimidate you, though! Handling complex numbers is just a matter of remembering that √(-1) = i. So, let's keep going and see how this unfolds. Remember, the beauty of the quadratic formula is that it guides you through each step, even when things get a little “complex.” This careful, methodical approach is what separates a correct solution from a near miss. Let's keep simplifying and unlock those roots!

Dealing with the Imaginary Unit

Now, let's tackle the square root of -12. Remember that $ ext{√}(-12)$ can be rewritten as $ ext{√}(12 imes -1)$, which is equal to $ ext{√}(12) imes ext{√}(-1)$. We know that $ ext{√}(-1)$ is the imaginary unit, denoted by i. So, we have:

$ ext{√}(-12) = ext{√}(12) * i$

We can further simplify $ ext{√}(12)$. Since 12 = 4 * 3, we have:

$ ext{√}(12) = ext{√}(4 * 3) = ext{√}(4) * ext{√}(3) = 2 ext{√}(3)$

Putting it all together, $ ext{√}(-12) = 2 ext{√}(3)i$. Now, we can substitute this back into our equation:

x = rac{6 extbf{±} 2 ext{√}(3)i}{2}

This step is crucial for solving quadratic equations with negative discriminants. Breaking down the square root and extracting i allows us to express the solutions in their proper complex form. It's a bit like decoding a secret message, where i is a key element of the code. Remember, complex numbers aren’t as “complex” as they seem once you get the hang of working with i. They follow specific rules, and mastering them opens up a whole new world of mathematical possibilities. The ability to handle imaginary units is a cornerstone of advanced algebra and calculus, so nailing it now will pay dividends later. Keep practicing this step, and you’ll find it becomes second nature. Let’s move on to the final simplification to reveal our solutions!

Final Simplification and the Solution

We're almost there! Our equation currently looks like this:

x = rac{6 extbf{±} 2 ext{√}(3)i}{2}

Notice that we can simplify this fraction by dividing both the real and imaginary parts by 2:

x = rac{6}{2} extbf{±} rac{2 ext{√}(3)i}{2}

This simplifies to:

x=3extbf±ext√(3)ix = 3 extbf{±} ext{√}(3)i

So, the solutions to the equation x2−6x+12=0x^2 - 6x + 12 = 0 are x=3+ext√(3)ix = 3 + ext{√}(3)i and x=3−ext√(3)ix = 3 - ext{√}(3)i. Looking back at our options, this matches option B.

Therefore, the correct solution is B. x=3extbf±iext√(3)x = 3 extbf{±} i ext{√}(3)

Congratulations, guys! We’ve successfully navigated through this quadratic equation, handling the complex solutions with confidence. Remember, the key to mastering these problems is understanding the quadratic formula, breaking down each step, and not being intimidated by the imaginary unit. Keep practicing, and you’ll become a quadratic equation-solving pro in no time!

Why Other Options Are Incorrect

Now, let’s briefly discuss why the other options are incorrect. This can help us understand common mistakes and reinforce the correct method.

  • A. x=6extbf±ext√(12)x = 6 extbf{±} ext{√}(12): This is incorrect because it doesn't correctly apply the quadratic formula. The formula requires dividing by 2a, and the square root portion was not handled correctly. Plus, it misses the imaginary unit, indicating a failure to recognize the negative discriminant.
  • C. x=−3extbf±iext√(3)x = -3 extbf{±} i ext{√}(3): This option gets the imaginary part right but makes a mistake with the real part. The sign of the real part should be positive 3, not negative 3. This could be a simple arithmetic error when applying the quadratic formula.
  • D. x=3extbf±2ix = 3 extbf{±} 2i: This option has the correct real part but incorrectly simplifies the imaginary part. The $ ext{√}(12)$ was likely simplified incorrectly, leading to the wrong coefficient for i. This highlights the importance of carefully simplifying square roots and complex numbers.

By understanding why these options are incorrect, we reinforce our understanding of the correct process and can avoid these pitfalls in the future. It's not just about getting the right answer; it's about understanding the why behind the answer.

Key Takeaways

Alright, guys, let's wrap up what we've learned today. Solving quadratic equations, especially those with complex solutions, can seem daunting, but here are the key takeaways to remember:

  1. The Quadratic Formula is Your Best Friend: When in doubt, use the quadratic formula. It's a reliable method for finding all solutions to a quadratic equation.
  2. Pay Attention to the Discriminant: The value inside the square root (b2−4acb^2 - 4ac) tells you a lot about the nature of the solutions. A negative discriminant means you'll have complex solutions.
  3. Master Complex Numbers: Don't be scared of the imaginary unit i. Remember that $ ext{√}(-1) = i$, and practice simplifying expressions involving i.
  4. Simplify, Simplify, Simplify: Always simplify your expressions as much as possible. This reduces the chance of making errors and makes the final answer clearer.
  5. Double-Check Your Work: It's easy to make a small arithmetic error, so always double-check your calculations, especially the signs and square root simplifications.

By keeping these points in mind, you'll be well-equipped to tackle any quadratic equation that comes your way. Keep practicing, and remember that every mistake is a learning opportunity. You've got this!

So, next time you encounter a quadratic equation like x2−6x+12=0x^2 - 6x + 12 = 0, you'll know exactly how to approach it. Remember the quadratic formula, the importance of simplifying square roots, and the magic of the imaginary unit. Keep practicing, and you’ll be solving these equations like a pro in no time. Until next time, keep those math skills sharp!