Solving Quadratic Equations: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of quadratic equations. Specifically, we'll tackle the equation $6x^2 - 3x + 6 = 0$. Our goal? To find the solutions and express them in the form $a \pm bi$. Don't worry if this sounds a bit intimidating; we'll break it down step by step to make sure everyone understands. This guide will walk you through the process, ensuring you not only solve the equation but also grasp the underlying concepts. So, grab your pencils and let's get started!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what a quadratic equation is. In its standard form, a quadratic equation looks like this: $ax^2 + bx + c = 0$. Here, 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to solve for. The key feature of a quadratic equation is the $x^2$ term, which means the highest power of the variable is 2. The solutions to these equations are also known as roots. These roots can be real numbers, or they can be complex numbers, which involve the imaginary unit 'i' (where $i = \sqrt{-1}$).

In our given equation, $6x^2 - 3x + 6 = 0$, we have $a = 6$, $b = -3$, and $c = 6$. Notice that the coefficients a, b, and c are all real numbers. When you come across these types of equations, understanding these components is vital for solving them correctly. We’re going to look for these solutions using the quadratic formula. In particular, we will use it to solve and identify complex solutions. The quadratic formula is a universal tool to find the roots of any quadratic equation, regardless of whether they are real or complex. Furthermore, we will simplify the solutions into the standard form. This is crucial for matching the format provided in the multiple-choice options, which is a common type of question on many standardized tests and exams. This process not only solves the equation but also deepens your understanding of complex numbers and their representation.

The Quadratic Formula: Your Best Friend

To solve our equation, we'll use the quadratic formula. This is a handy tool that provides a direct way to find the roots of any quadratic equation. The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Let's apply this formula to our equation, $6x^2 - 3x + 6 = 0$. Remember, $a = 6$, $b = -3$, and $c = 6$. Substituting these values into the formula, we get:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 6 \cdot 6}}{2 \cdot 6}$.

Now, we'll simplify this expression step by step. First, simplify the terms inside the square root and the numerator. Then we will be able to get a proper solution for the roots of the equation. This will ensure we perform the arithmetic operations accurately. The quadratic formula is an essential tool, so it's a good idea to know it off by heart, and be ready to use it in any situation.

Solving the Equation: Step-by-Step

Now, let's break down the calculations to find the solutions. Following the quadratic formula, we have $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 6 \cdot 6}}{2 \cdot 6}$. Let's simplify each part of this equation. The first step is to simplify the terms inside the square root, i.e., the discriminant: $(-3)^2 - 4 \cdot 6 \cdot 6$. This calculates to $9 - 144$, which equals $-135$. Since the discriminant is negative, we know that our solutions will involve complex numbers.

Next, simplify the rest of the equation to find the roots. The equation now looks like this: $x = \frac{3 \pm \sqrt{-135}}{12}$. Now we need to simplify the square root of -135. We can rewrite $\sqrt{-135}$ as $\sqrt{135} \cdot \sqrt{-1}$. The $\sqrt{-1}$ is equal to 'i', which is the imaginary unit. We can simplify $\sqrt{135}$ by factoring it. $135 = 9 \cdot 15$, so $\sqrt{135} = \sqrt{9} \cdot \sqrt{15} = 3 \sqrt{15}$. Therefore, $\sqrt{-135} = 3i\sqrt{15}$.

Substituting this back into the equation gives us $x = \frac{3 \pm 3i\sqrt{15}}{12}$. To simplify further, we can divide each term in the numerator by 3, resulting in $x = \frac{1 \pm i\sqrt{15}}{4}$.

So, the solutions in the form $a \pm bi$ are $\frac{1}{4} \pm \frac{\sqrt{15}}{4}i$. This solution perfectly matches one of the options in the multiple-choice question. This detailed process of simplification is crucial to make sure all parts of the calculation are accurate. By consistently applying the formula and simplifying correctly, you can solve any quadratic equation with confidence.

Matching the Solution to the Options

Now that we've found our solutions, let's match them with the given options:

A. $\frac{1}{4} \pm \frac{\sqrt{17}}{4} i$ B. $\frac{1}{4} \pm \frac{\sqrt{15}}{4} i$ C. $\frac{1}{2} \pm \frac{\sqrt{15}}{2} i$

D. (Not provided)

The solution we found is $\frac{1}{4} \pm \frac{\sqrt{15}}{4}i$, which perfectly matches option B. Thus, the correct answer is B. Recognizing the complex solutions and simplifying them to match the required format is a critical skill for this type of problem. Being able to solve and identify complex solutions not only shows that you understand the concepts, but also that you can correctly apply the formulas and manipulate the numbers. So, good job, everyone!

Conclusion

And there you have it! We've successfully solved the quadratic equation $6x^2 - 3x + 6 = 0$ and expressed the solutions in the form $a \pm bi$. We started with the quadratic formula, carefully substituted the values, simplified the expression, and finally matched our answer with the given options. Remember, practice is key. The more you work through these problems, the more comfortable you'll become with the concepts. Keep practicing, and you'll become a quadratic equation master in no time! Keep up the great work, and happy solving!