Solving $3x^2 - 2x - 6 = 0$ Using The Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions (also called roots or zeros) of any quadratic equation in the standard form of $ax^2 + bx + c = 0$. In this article, we will delve into the process of using the quadratic formula to solve the equation $3x^2 - 2x - 6 = 0$. This equation is a classic example that showcases the effectiveness and necessity of the quadratic formula when factoring or other simpler methods don't readily provide the solutions. We'll break down each step, ensuring a clear understanding of how to apply the formula and interpret the results. Understanding the nuances of the quadratic formula is crucial for anyone studying algebra, calculus, or related fields, as it forms a foundational concept for more advanced mathematical problem-solving. By mastering this technique, you'll be equipped to tackle a wide array of quadratic equations and gain a deeper appreciation for the elegance and utility of algebraic methods. This article aims to provide a comprehensive guide, suitable for both beginners and those looking to refresh their knowledge, with a focus on accuracy and clarity in each step of the solution.

The quadratic formula is expressed as:

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

Where:

• $x$ represents the solutions (roots) of the quadratic equation.
• $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

1. Identify the Coefficients

The first step in applying the quadratic formula is to correctly identify the coefficients $a$, $b$, and $c$ from the given quadratic equation. This is a crucial step because the accuracy of the subsequent calculations depends on these values. A mistake in identifying these coefficients can lead to incorrect solutions, so it's important to be meticulous in this initial phase. Let's take a closer look at how to extract these values from our equation, $3x^2 - 2x - 6 = 0$. By carefully comparing the given equation with the standard form $ax^2 + bx + c = 0$, we can clearly see the correspondence between the terms. This direct comparison method is a reliable way to avoid confusion, especially when dealing with equations that have negative coefficients or when the terms are not arranged in the standard order. Identifying the correct coefficients sets the stage for the accurate application of the quadratic formula, ensuring that we proceed with a solid foundation towards finding the roots of the equation. The ability to accurately identify coefficients is not only essential for solving quadratic equations but also for understanding the structure and properties of polynomial equations in general.

In the equation $3x^2 - 2x - 6 = 0$, we have:

• $a = 3$ (the coefficient of $x^2$)
• $b = -2$ (the coefficient of $x$)
• $c = -6$ (the constant term)

2. Substitute the Values into the Quadratic Formula

Once we have correctly identified the coefficients $a$, $b$, and $c$, the next step is to substitute these values into the quadratic formula. This process involves replacing the variables in the formula with their corresponding numerical values. This is a critical step, as it sets the stage for the arithmetic calculations that will lead us to the solutions of the equation. Care should be taken to ensure that the values are substituted correctly, paying close attention to signs (positive and negative) to avoid errors. A common mistake is mishandling negative signs, so double-checking this step is always a good practice. By accurately substituting the coefficients, we transform the abstract formula into a concrete expression that we can evaluate. This substitution is a mechanical process, but it requires precision to ensure the integrity of the solution. The careful substitution of values is a foundational skill in algebra, applicable not only to the quadratic formula but also to a wide range of mathematical formulas and equations. With the correct substitution, we are well on our way to solving the equation and uncovering the values of $x$ that satisfy it.

Now, we substitute $a = 3$, $b = -2$, and $c = -6$ into the quadratic formula:

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

3. Simplify the Expression

After substituting the values into the quadratic formula, the next crucial step is to simplify the resulting expression. This involves performing the arithmetic operations in the correct order, following the standard mathematical conventions such as the order of operations (PEMDAS/BODMAS). The simplification process is vital because it transforms the complex expression into a more manageable form, making it easier to isolate the variable and find the solutions. This step often involves dealing with square roots, fractions, and arithmetic operations with positive and negative numbers, so a thorough understanding of these concepts is essential. Accuracy in simplification is paramount; any error in this step can propagate through the rest of the solution, leading to an incorrect answer. Furthermore, simplifying the expression not only makes it easier to solve but also provides insights into the nature of the solutions, such as whether they are real or complex. Mastery of simplification techniques is a fundamental skill in mathematics, applicable across various domains beyond quadratic equations. It builds a strong foundation for more advanced algebraic manipulations and problem-solving strategies.

Let's simplify the expression step by step:

$$x = \frac{2 \pm \sqrt{4 + 72}}{6}$$

$$x = \frac{2 \pm \sqrt{76}}{6}$$

4. Calculate the Square Root

The next step in solving the quadratic equation after simplifying the expression is to calculate the square root. This often involves finding the principal square root of a number, which is the non-negative value that, when multiplied by itself, equals the number under the radical. In many cases, the number under the square root (the radicand) is not a perfect square, meaning its square root is an irrational number. In such instances, we either approximate the square root to a certain number of decimal places or simplify it by factoring out perfect square factors from the radicand. Calculating the square root accurately is essential because it directly impacts the final solutions of the equation. If the square root is a real number, it indicates that the quadratic equation has real solutions. Conversely, if the number under the square root is negative, it implies that the solutions are complex numbers. Thus, the square root calculation not only provides a numerical value but also gives insight into the nature of the roots. Proficiency in handling square roots is a valuable skill in algebra and is frequently used in various mathematical contexts, including geometry, trigonometry, and calculus.

We can simplify $\sqrt{76}$ by factoring out the largest perfect square factor, which is 4:

$$\sqrt{76} = \sqrt{4 \cdot 19} = 2\sqrt{19}$$

So the expression becomes:

$$x = \frac{2 \pm 2\sqrt{19}}{6}$$

5. Simplify the Fraction

After calculating the square root and substituting it back into the expression, the next crucial step is to simplify the fraction. This involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). Simplifying fractions is a fundamental skill in mathematics and is essential for expressing solutions in their most concise and understandable form. This process often requires identifying common factors in both the numerator and the denominator, which can involve factoring numbers or algebraic expressions. Accurate simplification of fractions is vital because it ensures that the solutions are presented in a clear and precise manner, avoiding any potential ambiguity. Moreover, it can also make subsequent calculations or interpretations easier to handle. Mastery of fraction simplification is a valuable asset in algebra and is frequently used in various mathematical contexts, including equation solving, calculus, and mathematical modeling. It is a core skill that builds a solid foundation for more advanced mathematical concepts and techniques.

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$$x = \frac{2(1 \pm \sqrt{19})}{2(3)}$$

$$x = \frac{1 \pm \sqrt{19}}{3}$$

6. Find the Two Solutions

The quadratic formula yields two possible solutions for $x$ because of the $\pm$ (plus or minus) sign in the formula. This indicates that there are typically two values of $x$ that will satisfy the quadratic equation. These two solutions arise from considering both the addition and subtraction of the square root term in the numerator. Finding both solutions is crucial for a complete answer to the problem. These solutions may be real and distinct, real and equal (when the discriminant is zero), or complex (when the discriminant is negative). The nature of these solutions provides valuable information about the quadratic equation and its corresponding graph. Understanding how to correctly identify and calculate both solutions is a fundamental aspect of solving quadratic equations. This skill is not only essential in algebra but also has applications in various fields, such as physics, engineering, and computer science, where quadratic equations are frequently encountered. Mastery of finding both solutions is key to a comprehensive understanding of quadratic equations and their applications.

Now, we find the two solutions:

$$x_1 = \frac{1 + \sqrt{19}}{3}$$

$$x_2 = \frac{1 - \sqrt{19}}{3}$$

7. Approximate the Solutions to the Nearest Hundredth

Since the problem requires rounding the answers to the nearest hundredth, we need to approximate the values of the square root and then perform the calculations. This step is crucial for providing solutions that are practical and easily interpretable, especially in real-world applications where precise values are often not necessary or feasible. Approximating to a specific decimal place ensures that the answers are within a reasonable range of accuracy while also making them simpler to use in further calculations or analyses. This process involves using either a calculator or manual methods to estimate the value of the square root and then performing the addition, subtraction, and division as indicated in the expression. Accuracy in approximation is important to maintain the integrity of the solution, so it's essential to use appropriate rounding rules and techniques. The ability to approximate solutions is a valuable skill in mathematics and science, as it allows for practical problem-solving and decision-making even when exact solutions are difficult or impossible to obtain. This step bridges the gap between theoretical solutions and their real-world applications.

Using a calculator, we find:

$$\sqrt{19} \approx 4.3589$$

Now, we can approximate the solutions:

$$x_1 = \frac{1 + 4.3589}{3} \approx \frac{5.3589}{3} \approx 1.7863$$

Rounding to the nearest hundredth, we get $x_1 \approx 1.79$.

$$x_2 = \frac{1 - 4.3589}{3} \approx \frac{-3.3589}{3} \approx -1.1196$$

Rounding to the nearest hundredth, we get $x_2 \approx -1.12$.

8. Final Answer

Therefore, the solutions to the equation $3x^2 - 2x - 6 = 0$, rounded to the nearest hundredth, are approximately $x \approx 1.79$ and $x \approx -1.12$. This final step consolidates all the previous calculations and approximations to present the solutions in a clear and concise manner. It is important to state the solutions explicitly, ensuring that the answer is easily understood and directly addresses the original problem. The final answer represents the values of $x$ that, when substituted back into the original equation, will make the equation true (or very close to true, given the rounding). Presenting the final answer accurately is a crucial aspect of mathematical problem-solving, as it demonstrates a clear understanding of the process and the results. This step is not just about providing the numerical values but also about communicating the solution effectively, which is a key skill in both academic and professional settings. The ability to arrive at and clearly state the final answer is a testament to the problem-solving proficiency and mathematical acumen of the solver.

In summary, using the quadratic formula to solve $3x^2 - 2x - 6 = 0$ involves identifying the coefficients, substituting them into the formula, simplifying the expression, calculating the square root, simplifying the fraction, finding the two solutions, and approximating those solutions to the nearest hundredth. The solutions are approximately $1.79$ and $-1.12$.