Solving 6x² + 5x - 4 = 0 A Comprehensive Guide

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In this article, we will delve into the process of finding the solutions to the quadratic equation 6x² + 5x - 4 = 0. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields, including mathematics, physics, engineering, and economics. Understanding how to solve them is a fundamental skill. We'll explore the methods used to solve this particular equation and verify which of the provided options are the correct solutions. To provide a comprehensive understanding, we'll cover the theory behind quadratic equations, the different methods to solve them, and apply these methods step-by-step to the given equation. This detailed approach ensures that readers grasp the concepts thoroughly and can apply them to similar problems.

Understanding Quadratic Equations

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A quadratic equation is a polynomial equation of the second degree, generally represented in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are also known as its roots or zeros, which represent the values of x that satisfy the equation. These roots correspond to the points where the parabola, described by the quadratic equation, intersects the x-axis. The nature and number of roots can be determined by the discriminant, which is given by the formula Δ = b² - 4ac. If Δ > 0, the equation has two distinct real roots; if Δ = 0, it has one real root (or two equal real roots); and if Δ < 0, it has no real roots (two complex roots). Understanding the discriminant is crucial for predicting the type of solutions we will encounter. In our specific case, 6x² + 5x - 4 = 0, we have a = 6, b = 5, and c = -4. We will use these coefficients to find the solutions and verify the given options. The solutions can be found using various methods, such as factoring, completing the square, or using the quadratic formula. Each method has its own advantages and disadvantages, depending on the specific equation. For example, factoring is often the quickest method when the equation can be easily factored, while the quadratic formula is a universal method that works for all quadratic equations. The choice of method often depends on the specific characteristics of the equation and the solver's preference. By mastering these methods, one can confidently solve a wide range of quadratic equations.

Methods for Solving Quadratic Equations

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There are several methods to solve quadratic equations, each with its own strengths and applications:

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. This method is most effective when the quadratic equation can be easily factored. The goal is to rewrite the equation in the form (px + q)(rx + s) = 0, where p, q, r, and s are constants. Once factored, the solutions can be found by setting each factor equal to zero and solving for x. Factoring relies on recognizing patterns and relationships between the coefficients of the quadratic equation. For example, in the equation x² + 5x + 6 = 0, we look for two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3, so the equation can be factored as (x + 2)(x + 3) = 0. Setting each factor to zero gives the solutions x = -2 and x = -3. However, not all quadratic equations can be easily factored, which is where other methods come into play. Factoring is often the quickest method when it works, but it requires some intuition and practice to become proficient. The success of factoring often depends on the solver's ability to recognize the specific patterns and relationships within the equation. Therefore, while it's a powerful tool, it's not always the most reliable for complex quadratic equations.

2. Completing the Square

Completing the square is a method that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved. This method involves manipulating the equation to create a square on one side, allowing us to take the square root and solve for x. The process involves adding and subtracting a constant to the equation to create the perfect square. Completing the square is a powerful method because it can be used to solve any quadratic equation, regardless of whether it can be easily factored. The general steps for completing the square involve dividing the equation by the leading coefficient (if it's not 1), moving the constant term to the other side of the equation, adding the square of half of the coefficient of x to both sides, and then rewriting the quadratic expression as a squared binomial. For example, to solve x² + 6x + 5 = 0, we would first move the constant term to get x² + 6x = -5. Then, we take half of the coefficient of x (which is 3), square it (which is 9), and add it to both sides: x² + 6x + 9 = -5 + 9. This simplifies to (x + 3)² = 4. Taking the square root of both sides gives x + 3 = ±2, leading to the solutions x = -1 and x = -5. While completing the square can be more involved than factoring, it provides a systematic approach that works for all quadratic equations. It also provides the basis for deriving the quadratic formula, which is a general solution for quadratic equations.

3. Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:

$$x = \frac{-b ± \sqrt{b² - 4ac}}{2a}$$

This formula directly provides the solutions by plugging in the coefficients a, b, and c from the quadratic equation. The quadratic formula is particularly useful when the equation is difficult to factor or when completing the square is cumbersome. It provides a straightforward and reliable way to find the roots of any quadratic equation. The term inside the square root, b² - 4ac, is the discriminant, which we mentioned earlier. The discriminant determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (or two equal real roots); and if it's negative, there are no real roots (two complex roots). The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0. This derivation demonstrates the formula's connection to the method of completing the square and provides a deeper understanding of its origins. Using the quadratic formula is a fundamental skill in algebra and is essential for solving a wide range of problems involving quadratic equations. Its universality and reliability make it a go-to method for many students and professionals alike.

Solving 6x² + 5x - 4 = 0

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Now, let's apply these methods to our equation, 6x² + 5x - 4 = 0. We'll start by attempting to factor the equation, then we'll use the quadratic formula to verify our solutions.

1. Factoring Method

To factor the quadratic equation 6x² + 5x - 4 = 0, we need to find two binomials that multiply to give the original equation. We look for two numbers that multiply to (6 * -4 = -24) and add up to 5. These numbers are 8 and -3. So, we can rewrite the middle term as 8x - 3x:

6x² + 8x - 3x - 4 = 0

Now, we factor by grouping:

2x(3x + 4) - 1(3x + 4) = 0

(2x - 1)(3x + 4) = 0

Setting each factor equal to zero gives us the solutions:

2x - 1 = 0 => x = 1/2

3x + 4 = 0 => x = -4/3

So, the solutions obtained by factoring are x = 1/2 and x = -4/3. Factoring involves reversing the process of expanding two binomials, which requires careful consideration of the coefficients. The ability to quickly and accurately factor quadratic equations is a valuable skill in algebra. While not all quadratic equations can be easily factored, those that can often yield solutions more quickly than other methods. The key is to look for the right combination of numbers that satisfy both the multiplication and addition conditions. In this case, identifying 8 and -3 as the numbers that multiply to -24 and add up to 5 was crucial for successfully factoring the equation. Factoring is not only a method for solving quadratic equations but also a fundamental concept in simplifying and manipulating algebraic expressions.

2. Quadratic Formula Method

Now, let's use the quadratic formula to solve 6x² + 5x - 4 = 0. Here, a = 6, b = 5, and c = -4. Plugging these values into the quadratic formula:

$$x = \frac{-5 ± \sqrt{5² - 4 * 6 * (-4)}}{2 * 6}$$

$$x = \frac{-5 ± \sqrt{25 + 96}}{12}$$

$$x = \frac{-5 ± \sqrt{121}}{12}$$

$$x = \frac{-5 ± 11}{12}$$

This gives us two solutions:

$$x = \frac{-5 + 11}{12} = \frac{6}{12} = \frac{1}{2}$$

$$x = \frac{-5 - 11}{12} = \frac{-16}{12} = -\frac{4}{3}$$

The solutions obtained using the quadratic formula are x = 1/2 and x = -4/3, which confirms the solutions we found by factoring. The quadratic formula provides a direct and reliable way to find the solutions, especially when factoring is not straightforward. It's a powerful tool that works for any quadratic equation, regardless of its complexity. The formula's structure reflects the fundamental properties of quadratic equations and their solutions. The discriminant, b² - 4ac, plays a crucial role in determining the nature of the roots, as we discussed earlier. The quadratic formula is not just a mathematical tool but also a fundamental concept in algebra, providing a complete solution to quadratic equations. Its derivation and application demonstrate the interconnectedness of various algebraic techniques and concepts.

Checking the Options

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Now, let's check the given options to see which ones match our solutions:

A. x = -1/3 B. x = -4/3 C. x = -2 D. x = 4 E. x = 1/2 F. x = 1/3

Comparing these options with our solutions, x = 1/2 and x = -4/3, we find that options B and E are the correct solutions.

Conclusion

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In conclusion, by solving the quadratic equation 6x² + 5x - 4 = 0 using both factoring and the quadratic formula, we found the solutions to be x = 1/2 and x = -4/3. Therefore, the correct options are B and E. Understanding and mastering the methods for solving quadratic equations is essential for various mathematical and real-world applications. The ability to solve quadratic equations is a fundamental skill in algebra, with applications in a wide range of fields, including physics, engineering, economics, and computer science. The methods we've discussed, factoring, completing the square, and using the quadratic formula, provide a comprehensive toolkit for tackling any quadratic equation. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and the solver's preference. By practicing and applying these methods, one can develop a deep understanding of quadratic equations and their solutions. The study of quadratic equations also lays the foundation for more advanced topics in mathematics, such as polynomial equations and calculus. Therefore, mastering this topic is crucial for students pursuing further studies in STEM fields.