Unlock The Solution To 3x^2-x+2=0

When faced with a quadratic equation like $3x^2-x+2=0$, the first thing that often comes to mind is, "How do I even begin to solve this?" Well, you're in luck! We're going to dive deep into solving this particular equation, breaking down the process step-by-step so that it's not just understandable, but crystal clear. This isn't just about getting an answer; it's about understanding the methodology behind solving quadratic equations. We'll explore the tools at our disposal, primarily the quadratic formula, and see how it elegantly untangles the complexity of these equations. So, grab a pen and paper, and let's get ready to unravel the mystery of $3x^2-x+2=0$. We'll ensure that by the end of this discussion, you'll feel confident tackling similar problems. Remember, the journey of a thousand miles begins with a single step, and understanding quadratic equations is a crucial step in your mathematical journey.

Understanding the Quadratic Formula: Your Mathematical Toolkit

The quadratic formula is, without a doubt, one of the most powerful tools in a mathematician's arsenal when dealing with quadratic equations. A quadratic equation is generally expressed in the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are coefficients, and 'x' is the variable we're trying to solve for. The quadratic formula itself is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula is derived from the process of completing the square on the general quadratic equation and provides a direct route to finding the values of 'x' that satisfy the equation, known as the roots or solutions. It's incredibly versatile because it works for any quadratic equation, whether the roots are real, complex, distinct, or repeated. Understanding the components of the formula is key: '-b' is the negative of the coefficient of the linear term, '$b^2 - 4ac$' is the discriminant (which tells us about the nature of the roots), and '2a' is twice the coefficient of the squared term. When applying the formula, precision is paramount. Substituting the correct values for a, b, and c is the first critical step. Any small error in this substitution can lead to an incorrect final answer. Therefore, it's always a good practice to identify these coefficients explicitly before plugging them into the formula. The discriminant, $b^2 - 4ac$, is particularly interesting. If it's positive, you get two distinct real roots. If it's zero, you get one repeated real root. If it's negative, you get two complex conjugate roots. This insight into the nature of the roots before calculating them is one of the many beauties of the quadratic formula. Mastering this formula is not just about memorization; it's about developing a deep understanding of its application and significance in solving a wide range of algebraic problems.

Step-by-Step Solution for $3x^2-x+2=0$

Now, let's apply our trusted quadratic formula to the specific equation: $3x^2-x+2=0$. The first step in solving any quadratic equation using the formula is to identify the coefficients a, b, and c. In our equation, we can see that: $a = 3$ (the coefficient of $x^2$), $b = -1$ (the coefficient of x), and $c = 2$ (the constant term). It's crucial to pay attention to the signs; the negative sign in front of the 'x' means that b is -1, not just 1. Once we have these values clearly identified, we can substitute them into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Plugging in our values, we get: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(2)}}{2(3)}$. Let's simplify this expression step by step. The $-(-1)$ in the numerator becomes $+1$. The term inside the square root, the discriminant, is $(-1)^2 - 4(3)(2)$. Squaring -1 gives us 1. Then, $4(3)(2) = 24$. So, the discriminant is $1 - 24 = -23$. The denominator is $2(3) = 6$. Therefore, our equation simplifies to: $x = \frac{1 \pm \sqrt{-23}}{6}$. Now, we encounter a negative number under the square root. In the realm of real numbers, the square root of a negative number is undefined. However, in the realm of complex numbers, we can express $\sqrt{-23}$ as $\sqrt{23}i$, where 'i' is the imaginary unit, defined as $\sqrt{-1}$. So, the solutions for x are $x = \frac{1 \pm \sqrt{23}i}{6}$. This means we have two complex solutions: $x_1 = \frac{1 + \sqrt{23}i}{6}$ and $x_2 = \frac{1 - \sqrt{23}i}{6}$. These are the precise solutions to the quadratic equation $3x^2-x+2=0$. The process highlights the importance of careful substitution and simplification. Even when the discriminant is negative, the quadratic formula provides a clear path to the complex solutions.

Analyzing the Options Provided

When presented with multiple-choice options for solving an equation, it's essential to not only find the correct answer but also to understand why the other options are incorrect. This reinforces your understanding and helps you spot common pitfalls. Let's examine the options provided for solving $3x^2-x+2=0$: A. z=rac{1}{8} oon rac{42}{6}, B. z=rac{1}{8} oon rac{42}{8}, C. z=rac{-1+\sqrt{2}}{8}, D. z=rac{-1+\sqrt{2} 2}{6}, E. z=rac{-1+\sqrt{2}}{6}. Our calculated solutions are $x = \frac{1 \pm \sqrt{-23}}{6}$. Notice that none of the options directly match our result. This could indicate a typo in the options or a misunderstanding of the problem's context (perhaps the question intended to ask for a different equation or a different type of solution). However, assuming the equation $3x^2-x+2=0$ is correct, let's analyze why the options deviate. Options A and B involve a $\frac{1}{8}$ term and $\frac{42}{6}$ or $\frac{42}{8}$. This structure doesn't align with the standard quadratic formula output for our equation. The denominator '8' in options A, B, and C is particularly suspicious, as the quadratic formula for our equation yields a denominator of $2a = 2(3) = 6$. Options C, D, and E all have denominators of 6 or 8, which is a slightly better fit for options D and E, but the numerators are where the significant differences lie. Option C, z=rac{-1+\sqrt{2}}{8}, has a -1 in the numerator and $\sqrt{2}$, which doesn't match our $-b = -(-1) = 1$ and $\sqrt{b^2-4ac} = \sqrt{-23}$. Option D, z=rac{-1+\sqrt{2} 2}{6}, has a $-1$ and $\sqrt{2}$, which is also incorrect. Option E, z=rac{-1+\sqrt{2}}{6}, has a $-1$ and $\sqrt{2}$ in the numerator, and a denominator of 6. While the denominator matches, the numerator does not correspond to our calculation of $1 \pm \sqrt{-23}$. It's possible the original problem intended to be something like $x^2 + x - 1 = 0$ or a similar equation that would yield results closer to these options. For instance, if the equation was $6x^2+x-1=0$, the quadratic formula would give $x = \frac{-1 oon \sqrt{1^2 - 4(6)(-1)}}{2(6)} = \frac{-1 oon \sqrt{1+24}}{12} = \frac{-1 oon \sqrt{25}}{12} = \frac{-1 oon 5}{12}$, giving $x = \frac{4}{12} = \frac{1}{3}$ or x = \frac{-6}{12} = -rac{1}{2}. This still doesn't match. The presence of $\sqrt{2}$ in options C, D, and E suggests that perhaps the discriminant was intended to be 2. Let's assume, for a moment, that the discriminant $b^2-4ac$ was meant to be 2. Then we'd have $x = \frac{-b oon \sqrt{2}}{2a}$. If we also assume $a=6$ and $b=-1$, then $x = \frac{1 oon \sqrt{2}}{12}$. This is still not quite right. The given options seem to be based on a different quadratic equation entirely, or there are significant errors in the options themselves. However, based on the equation $3x^2-x+2=0$ and the standard quadratic formula, our derived complex solutions $x = \frac{1 \pm \sqrt{-23}}{6}$ are the correct ones. If forced to choose the closest form, one might look for a denominator of 6, but the numerators in options C, D, and E are fundamentally different from our calculated $1 oon \sqrt{-23}$.

Conclusion: The Correct Path to Solution

In summary, when tackling the quadratic equation $3x^2-x+2=0$, the most reliable and systematic approach is to utilize the quadratic formula. We identified our coefficients as $a=3$, $b=-1$, and $c=2$. Substituting these into the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ led us to $x = \frac{1 \pm \sqrt{-23}}{6}$. This results in two complex conjugate solutions: $x = \frac{1 + \sqrt{-23}}{6}$ and $x = \frac{1 - \sqrt{-23}}{6}$, which can also be written using the imaginary unit 'i' as $x = \frac{1 \pm i\sqrt{23}}{6}$. It's important to note that the multiple-choice options provided do not seem to align with the correct solution for this specific equation. This could stem from a typo in the equation itself or in the options. However, the methodology we've followed is sound and universally applicable. Always remember to carefully identify your coefficients, substitute them accurately into the quadratic formula, and simplify step-by-step. Paying close attention to the discriminant ($b^2 - 4ac$) will also give you valuable insight into the nature of the roots—whether they are real and distinct, real and repeated, or complex. While the provided options might be misleading, the journey of solving $3x^2-x+2=0$ using the quadratic formula has reinforced the power and elegance of this mathematical tool. Keep practicing, and you'll master these concepts in no time!

For further exploration into the fascinating world of quadratic equations and algebraic solutions, you can visit Wolfram MathWorld for in-depth mathematical resources and Khan Academy for comprehensive learning materials.