Solving 5x = 6x^2 - 3 Using The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \ne 0$. This article delves into using the quadratic formula to solve the equation $5x = 6x^2 - 3$. We will walk through the steps to rearrange the equation into the standard quadratic form, identify the coefficients, and apply the quadratic formula to find the values of $x$. Understanding how to use the quadratic formula is a fundamental skill in algebra, as it provides a reliable method for finding the roots (or solutions) of any quadratic equation, regardless of whether it can be easily factored. This comprehensive guide will help you grasp the nuances of the process, ensuring you can confidently tackle similar problems in the future. We will also discuss common pitfalls and strategies to avoid them, thereby enhancing your problem-solving skills in mathematics. The quadratic formula is not just a mathematical tool; it's a gateway to understanding more complex concepts in higher mathematics and real-world applications.

Rearranging the Equation

To begin, the given equation $5x = 6x^2 - 3$ needs to be rearranged into the standard quadratic form, which is $ax^2 + bx + c = 0$. This rearrangement is crucial because the coefficients $a$, $b$, and $c$ are directly used in the quadratic formula. Start by moving all terms to one side of the equation to set it equal to zero. Subtract $5x$ from both sides of the equation $5x = 6x^2 - 3$ to get $0 = 6x^2 - 5x - 3$. Now, we can rewrite this as $6x^2 - 5x - 3 = 0$. This form clearly shows the quadratic expression set to zero, making it ready for further analysis. Rearranging the equation correctly is a foundational step, and any error here will propagate through the rest of the solution. Double-checking this step is always a good practice to ensure accuracy. Recognizing the standard form of a quadratic equation is also vital for identifying the coefficients correctly, which we will discuss in the next section. The process of rearranging not only prepares the equation for solving but also helps in visualizing the structure of the quadratic expression.

Identifying the Coefficients

Once the equation is in the standard form $ax^2 + bx + c = 0$, we can identify the coefficients $a$, $b$, and $c$. In our rearranged equation, $6x^2 - 5x - 3 = 0$, the coefficients are as follows: $a$ is the coefficient of the $x^2$ term, which is 6; $b$ is the coefficient of the $x$ term, which is -5; and $c$ is the constant term, which is -3. It is extremely important to correctly identify these coefficients because they are directly substituted into the quadratic formula. A mistake in identifying even one coefficient will lead to an incorrect solution. Pay close attention to the signs of the coefficients, especially the negative signs, as they are a common source of errors. For instance, in our equation, $b$ is -5, not 5, and $c$ is -3, not 3. These signs are crucial in the subsequent steps. Understanding the role of each coefficient is also important. The coefficient $a$ determines the parabola's direction (whether it opens upwards or downwards) and its width, $b$ affects the parabola's position in the coordinate plane, and $c$ represents the y-intercept. With the coefficients correctly identified, we are now ready to apply the quadratic formula.

Applying the Quadratic Formula

The quadratic formula is given by:

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

This formula provides the solutions for $x$ in the quadratic equation $ax^2 + bx + c = 0$. We have already identified the coefficients $a = 6$, $b = -5$, and $c = -3$. Now, we substitute these values into the quadratic formula:

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(6)(-3)}}{2(6)}$$

Let's simplify this step by step. First, $-(-5)$ becomes 5. Next, $(-5)^2$ is 25, and $4(6)(-3)$ is -72. So, the expression under the square root becomes $25 - (-72)$, which is $25 + 72 = 97$. The denominator is $2(6) = 12$. Thus, the equation becomes:

$$x = \frac{5 \pm \sqrt{97}}{12}$$

The $\pm$ sign indicates that there are two possible solutions for $x$: one with addition and one with subtraction. The term inside the square root, $b^2 - 4ac$, is called the discriminant, and it determines the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. In our case, the discriminant is 97, which is positive, so we have two distinct real roots. The expression $\frac{5 \pm \sqrt{97}}{12}$ represents these two solutions in a concise form. Calculating the square root of 97 gives us an approximate value, but leaving it in this form provides an exact solution. We can now see that the solution matches one of the given options.

Final Answer

The values of $x$ obtained by applying the quadratic formula are:

$$x = \frac{5 \pm \sqrt{97}}{12}$$

Comparing this result with the given options, we can see that it matches option A. Therefore, the correct answer is:

A. $\frac{5 \pm \sqrt{97}}{12}$

This detailed step-by-step solution demonstrates the application of the quadratic formula to solve a quadratic equation. By rearranging the equation into standard form, identifying the coefficients, and substituting them into the formula, we have found the values of $x$. This method is applicable to any quadratic equation and is a fundamental technique in algebra. The ability to solve quadratic equations is crucial for various mathematical and real-world applications, from physics and engineering to economics and computer science. Understanding the quadratic formula not only helps in finding solutions but also provides insights into the nature of the roots and the behavior of quadratic functions. By mastering this technique, you can confidently approach and solve a wide range of quadratic equation problems.

In summary, solving the quadratic equation $5x = 6x^2 - 3$ using the quadratic formula involves several critical steps. First, the equation must be rearranged into the standard form $ax^2 + bx + c = 0$, which in this case becomes $6x^2 - 5x - 3 = 0$. Next, the coefficients $a$, $b$, and $c$ are identified as 6, -5, and -3, respectively. These values are then substituted into the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which simplifies to $x = \frac{5 \pm \sqrt{(-5)^2 - 4(6)(-3)}}{2(6)}$. After performing the calculations, we arrive at the solutions $x = \frac{5 \pm \sqrt{97}}{12}$. This result matches option A, confirming it as the correct answer. The process highlights the importance of careful rearrangement, accurate identification of coefficients, and precise application of the quadratic formula. Mastering these steps ensures the ability to solve a wide variety of quadratic equations efficiently and accurately. The quadratic formula is a cornerstone of algebraic problem-solving, providing a reliable method for finding roots that are not easily factored. This skill is essential for further studies in mathematics and its applications in various fields. Understanding and applying the quadratic formula is a valuable asset in any mathematical toolkit.