Solve $3x^2 - 4x + 6 = 2$: A Step-by-Step Guide

$$

Hey guys! Today, we're going to dive into solving a quadratic equation. Specifically, we'll tackle the equation $3x^2 - 4x + 6 = 2$. Don't worry, it's not as intimidating as it looks! We'll break it down step by step, making sure everyone can follow along. So, let's get started and find those solutions!

Understanding Quadratic Equations

Before we jump into the specifics of this problem, let's quickly recap what quadratic equations are all about. A quadratic equation is essentially a polynomial equation of the second degree. This means the highest power of the variable (in our case, x) is 2. The general form of a quadratic equation is given by:

$ax^2 + bx + c = 0$

Where a, b, and c are constants, and a is not equal to zero (otherwise, it wouldn't be a quadratic equation!). These constants determine the shape and position of the parabola that represents the equation when graphed. The solutions to the quadratic equation, also known as the roots or zeros, are the points where the parabola intersects the x-axis. These solutions can be real or complex numbers, depending on the specific equation.

Key Characteristics of Quadratic Equations:

• They always have two solutions (roots), which might be real or complex, and they might be the same (repeated roots).
• The graph of a quadratic equation is a parabola, a U-shaped curve.
• The coefficients a, b, and c influence the parabola's shape and position. For instance, 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0).

Knowing these basics is crucial because it helps us understand the context of the problem we are trying to solve. Now that we have a refresher, let’s jump back to our equation and see how we can find its solutions.

Setting Up the Equation

Alright, let’s get our hands dirty with the given equation: $3x^2 - 4x + 6 = 2$. The first thing we need to do is to put it into the standard form, which, as we discussed earlier, is $ax^2 + bx + c = 0$. To do this, we'll subtract 2 from both sides of the equation. This ensures that one side of the equation equals zero, which is essential for solving quadratic equations using methods like factoring or the quadratic formula.

So, here’s the step:

$3x^2 - 4x + 6 - 2 = 2 - 2$

Simplifying this, we get:

$3x^2 - 4x + 4 = 0$

Now, our equation is in the standard form, where a = 3, b = -4, and c = 4. Identifying these coefficients is a crucial step because they are the key ingredients we'll use in the quadratic formula. Trust me, keeping track of these values will make the rest of the process much smoother. We’ve got the equation set up perfectly, so let’s move on to the exciting part – actually solving it!

Applying the Quadratic Formula

Now comes the hero of our story – the quadratic formula! This formula is a universal tool for solving any quadratic equation, no matter how messy it looks. It's derived from the method of completing the square and provides us with a straightforward way to find the solutions. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Remember those coefficients a, b, and c we identified earlier? This is where they come into play. We're going to plug them into this formula and see what we get. In our case, we have a = 3, b = -4, and c = 4. Let’s substitute these values into the formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(4)}}{2(3)}$

Now, let’s simplify this step by step. First, we'll deal with the terms inside the square root and the other simpler operations:

$x = \frac{4 \pm \sqrt{16 - 48}}{6}$

Next, we simplify the expression under the square root:

$x = \frac{4 \pm \sqrt{-32}}{6}$

Notice that we have a negative number under the square root. This tells us that the solutions will be complex numbers, involving the imaginary unit i, where $i = \sqrt{-1}$. This is perfectly normal, and it just means our solutions will have both real and imaginary parts. Now, let's tackle that square root and see what we get!

Simplifying the Solution

Alright, we’re at the point where we need to simplify the expression $x = \frac{4 \pm \sqrt{-32}}{6}$. Remember, we have that pesky negative sign under the square root, which means we're dealing with imaginary numbers. No sweat, though! We can handle this by factoring out a -1 and using the definition of the imaginary unit, $i = \sqrt{-1}$.

First, let's rewrite $\sqrt{-32}$:

$\sqrt{-32} = \sqrt{32 \cdot -1} = \sqrt{32} \cdot \sqrt{-1} = \sqrt{32}i$

Now, we need to simplify $\sqrt{32}$. We can do this by finding the largest perfect square that divides 32. That would be 16, since 32 = 16 * 2. So, we can rewrite $\sqrt{32}$ as:

$\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$

Putting it all together, we have:

$\sqrt{-32} = 4\sqrt{2}i$

Now, let's plug this back into our equation:

$x = \frac{4 \pm 4\sqrt{2}i}{6}$

We can simplify this further by dividing both the real and imaginary parts by the greatest common divisor, which is 2:

$x = \frac{2 \pm 2\sqrt{2}i}{3}$

And there we have it! We've simplified the solutions as much as possible. This gives us two complex solutions for x. Now, let’s see how our solution matches up with the given options.

Matching with the Options

Okay, so we've arrived at the solutions:

$x = \frac{2 \pm 2\sqrt{2}i}{3}$

Now, let's compare this with the options provided:

A. $x=\frac{8 \pm \sqrt{2} i}{3}$ B. $x=\frac{3 \pm \sqrt{8} i}{2}$ C. $x=\frac{2 \pm 2 i \sqrt{2}}{3}$ D. $x=\frac{1 \pm \sqrt{6} i}{2}$ E. $x=\frac{2 \pm \sqrt{6} i}{3}$

Looking at the options, we can clearly see that our solution matches option C:

C. $x=\frac{2 \pm 2 i \sqrt{2}}{3}$

So, option C is the correct answer! We've successfully solved the quadratic equation and found the solutions that match one of the given choices. Awesome job, guys! You’ve navigated through the complexities of the quadratic formula and imaginary numbers like pros. Pat yourselves on the back!

Conclusion

In this article, we tackled the quadratic equation $3x^2 - 4x + 6 = 2$. We walked through the process of putting the equation into standard form, applying the quadratic formula, simplifying the solutions, and finally, matching our result with the provided options. Remember, the key to solving quadratic equations lies in understanding the standard form, knowing the quadratic formula, and being careful with the arithmetic, especially when dealing with square roots and imaginary numbers.

Solving quadratic equations is a fundamental skill in algebra, and mastering it opens the door to more advanced topics in mathematics and beyond. Whether you're a student prepping for an exam or just someone who enjoys mathematical challenges, I hope this step-by-step guide has been helpful. Keep practicing, and you’ll become a quadratic equation-solving whiz in no time! Keep up the excellent work, everyone! And remember, math can be fun when you break it down step by step.