Solving Quadratic Equations A Step By Step Guide

In this comprehensive guide, we will delve into the step-by-step process of solving the given equation, which involves algebraic manipulation and quadratic equation solving techniques. Our goal is to find the value(s) of x that satisfy the equation. The given equation presents a rational expression, which requires careful handling to avoid extraneous solutions. To start, we need to isolate the variable x and simplify the equation into a more manageable form. This often involves multiplying both sides by a common denominator, combining like terms, and rearranging the terms to achieve a standard quadratic equation format. This format allows us to apply standard methods such as factoring, completing the square, or using the quadratic formula to find the solutions. These methods provide a systematic approach to solve quadratic equations, ensuring accuracy and efficiency. Understanding these techniques is essential for solving various mathematical problems, not just within algebra but also in calculus and other advanced mathematical fields. Additionally, verifying the solutions by substituting them back into the original equation is a crucial step to confirm their validity and avoid any errors made during the solving process. The quadratic formula, in particular, is a powerful tool for solving any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are coefficients. By applying this formula, we can systematically find the roots of the equation. This process also reinforces the importance of understanding the underlying algebraic principles and the careful application of mathematical techniques. The ability to solve equations is a foundational skill in mathematics, with applications spanning various fields such as physics, engineering, and computer science. Therefore, mastering these techniques is crucial for anyone pursuing studies or careers in these areas.

Step-by-Step Solution

To solve the equation:

$$\frac{-5 x^2-4}{x-4}=\frac{-15 x}{x-4}+4$$

First, we will eliminate the fractions by multiplying both sides of the equation by (x - 4). This step is crucial for simplifying the equation and making it easier to solve. However, we must note that x cannot be equal to 4, as this would make the denominator zero, resulting in an undefined expression. By identifying this restriction early on, we can avoid potential extraneous solutions later in the solving process. Multiplying both sides by (x - 4) allows us to clear the fractions and obtain a more manageable equation, typically a polynomial equation. This process is a fundamental technique in algebra for solving equations involving rational expressions. It is essential to perform this step carefully, ensuring that each term is multiplied correctly to avoid errors. By doing so, we transform the original equation into a simpler form that can be further manipulated to isolate the variable x. This approach is widely used in various algebraic problems and is a key skill for anyone studying mathematics or related fields. The resulting polynomial equation can then be solved using various techniques, such as factoring, completing the square, or applying the quadratic formula. Each of these methods has its advantages and is suitable for different types of quadratic equations. Therefore, understanding these techniques and knowing when to apply them is crucial for solving a wide range of mathematical problems.

$$(x-4) \cdot \frac{-5 x^2-4}{x-4} = (x-4) \cdot \left(\frac{-15 x}{x-4}+4\right)$$

This simplifies to:

$$-5x^2 - 4 = -15x + 4(x - 4)$$

Next, we distribute the 4 on the right side:

$$-5x^2 - 4 = -15x + 4x - 16$$

Combine like terms on the right side:

$$-5x^2 - 4 = -11x - 16$$

Now, move all terms to one side to set the equation to zero:

$$-5x^2 + 11x + 12 = 0$$

Multiply the entire equation by -1 to make the leading coefficient positive. This step is optional but can make factoring or applying the quadratic formula easier.

$$5x^2 - 11x - 12 = 0$$

Now, we solve this quadratic equation. We can use the quadratic formula:

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

Where a = 5, b = -11, and c = -12.

Substitute the values:

$$x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(5)(-12)}}{2(5)}$$

$$x = \frac{11 \pm \sqrt{121 + 240}}{10}$$

$$x = \frac{11 \pm \sqrt{361}}{10}$$

$$x = \frac{11 \pm 19}{10}$$

So we have two possible solutions:

$$x_1 = \frac{11 + 19}{10} = \frac{30}{10} = 3$$

$$x_2 = \frac{11 - 19}{10} = \frac{-8}{10} = -\frac{4}{5}$$

Therefore, the solutions are x = 3 and x = -4/5.

Verification of Solutions

Verifying the solutions is a crucial step in solving equations, especially those involving rational expressions. This process ensures that the values obtained are indeed valid solutions and not extraneous roots introduced during the simplification steps. To verify the solutions, we substitute each value back into the original equation and check if both sides of the equation are equal. This step helps identify any errors made during the algebraic manipulation and confirms the accuracy of the solutions. In cases where the original equation involves denominators that could become zero for certain values of x, such as in the given equation, this verification step is particularly important. Substituting the solutions back into the original equation allows us to check if any of the denominators become zero, which would make the solution extraneous. This is because division by zero is undefined in mathematics, and such values cannot be valid solutions. Therefore, the verification step is not just a formality but an essential part of the problem-solving process, ensuring that the final answer is correct and complete. By taking the time to verify the solutions, we can have confidence in our results and avoid potential errors. This practice also reinforces the understanding of the underlying concepts and the importance of careful algebraic manipulation.

Let's verify the solutions by substituting them back into the original equation:

For $x = 3$:

$$\frac{-5(3)^2 - 4}{3 - 4} = \frac{-15(3)}{3 - 4} + 4$$

$$\frac{-5(9) - 4}{-1} = \frac{-45}{-1} + 4$$

$$\frac{-45 - 4}{-1} = 45 + 4$$

$$\frac{-49}{-1} = 49$$

$$49 = 49$$

This solution is valid.

For $x = -\frac{4}{5}$:

$$\frac{-5(-\frac{4}{5})^2 - 4}{-\frac{4}{5} - 4} = \frac{-15(-\frac{4}{5})}{-\frac{4}{5} - 4} + 4$$

$$\frac{-5(\frac{16}{25}) - 4}{-\frac{4}{5} - \frac{20}{5}} = \frac{12}{-\frac{4}{5} - \frac{20}{5}} + 4$$

$$\frac{-\frac{16}{5} - 4}{-\frac{24}{5}} = \frac{12}{-\frac{24}{5}} + 4$$

$$\frac{-\frac{16}{5} - \frac{20}{5}}{-\frac{24}{5}} = 12 \cdot -\frac{5}{24} + 4$$

$$\frac{-\frac{36}{5}}{-\frac{24}{5}} = -\frac{5}{2} + 4$$

$$\frac{36}{5} \cdot \frac{5}{24} = \frac{-5}{2} + \frac{8}{2}$$

$$\frac{36}{24} = \frac{3}{2}$$

$$\frac{3}{2} = \frac{3}{2}$$

This solution is also valid.

Final Answer

Therefore, the solutions to the equation are:

x = 3 and x = -4/5

The correct answer is B. x = 4/5, -3

Conclusion

In conclusion, the process of solving equations involves several key steps, including simplifying the equation, isolating the variable, and verifying the solutions. Understanding these steps is crucial for solving a wide range of mathematical problems. The given equation, a rational expression, required careful handling to avoid extraneous solutions. By multiplying both sides by the common denominator, we eliminated the fractions and simplified the equation into a standard quadratic form. This allowed us to apply the quadratic formula, a powerful tool for solving any quadratic equation. The quadratic formula provides a systematic way to find the roots of the equation, ensuring accuracy and efficiency. Furthermore, verifying the solutions by substituting them back into the original equation is an essential step to confirm their validity. This process helps identify any errors made during the algebraic manipulation and ensures that the final answer is correct. The ability to solve equations is a foundational skill in mathematics, with applications spanning various fields such as physics, engineering, and computer science. Therefore, mastering these techniques is crucial for anyone pursuing studies or careers in these areas. The step-by-step solution provided in this guide demonstrates the importance of careful algebraic manipulation and the application of appropriate mathematical tools. By following these steps, one can confidently solve similar equations and gain a deeper understanding of algebraic concepts. The use of examples and detailed explanations helps to reinforce the learning process and make the concepts more accessible. Ultimately, the goal is to equip students and learners with the skills and knowledge necessary to tackle complex mathematical problems with confidence and accuracy. This guide serves as a valuable resource for anyone seeking to improve their problem-solving abilities and deepen their understanding of mathematics.