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

Unlocking the roots of a quadratic equation is a fundamental skill in algebra. In this article, we will delve into the equation 6x² + 5x - 4 = 0, systematically exploring the methods to find its solutions. We'll not only identify the correct answers but also provide a step-by-step guide, ensuring a clear understanding of the underlying principles. Whether you're a student grappling with quadratic equations or simply seeking to refresh your algebra knowledge, this guide offers a comprehensive approach to mastering this essential concept. Let's embark on this mathematical journey together, demystifying the process of solving quadratic equations.

H2: Understanding Quadratic Equations

Before we dive into solving the specific equation, let's establish a solid foundation by understanding what quadratic equations are and the standard methods for solving them. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (typically 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola represented by the quadratic equation intersects the x-axis.

There are several methods for solving quadratic equations, each with its strengths and weaknesses. The most common methods include:

1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It's often the quickest method when the equation can be easily factored.
2. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of whether it can be factored easily. It's a powerful tool and a guaranteed method for finding the roots.
3. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial, which can then be easily solved.

In this article, we will primarily focus on factoring and the quadratic formula to solve the equation 6x² + 5x - 4 = 0. Understanding these methods will equip you with the skills to tackle a wide range of quadratic equations.

H2: Solving 6x² + 5x - 4 = 0 by Factoring

Factoring is a powerful technique for solving quadratic equations, and it's often the first method to try if the equation appears factorable. The goal of factoring is to rewrite the quadratic expression as a product of two linear expressions. To factor the quadratic equation 6x² + 5x - 4 = 0, we need to find two binomials that, when multiplied together, result in the original quadratic expression.

The process involves finding two numbers that multiply to give the product of the leading coefficient (6) and the constant term (-4), which is -24, and that add up to the middle coefficient (5). These two numbers are 8 and -3 because 8 * -3 = -24 and 8 + (-3) = 5. Now we can rewrite the middle term (5x) using these two numbers:

6x² + 5x - 4 = 6x² + 8x - 3x - 4

Next, we factor by grouping. We group the first two terms and the last two terms:

(6x² + 8x) + (-3x - 4)

Now, we factor out the greatest common factor (GCF) from each group:

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

Notice that both terms now have a common factor of (3x + 4). We can factor this out:

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

So, the factored form of the quadratic equation is (3x + 4)(2x - 1) = 0. To find the solutions, we set each factor equal to zero:

3x + 4 = 0 or 2x - 1 = 0

Solving these linear equations gives us the solutions:

3x = -4 => x = -4/3

2x = 1 => x = 1/2

Therefore, the solutions to the quadratic equation 6x² + 5x - 4 = 0, obtained through factoring, are x = -4/3 and x = 1/2.

H2: Solving 6x² + 5x - 4 = 0 Using the Quadratic Formula

While factoring is an efficient method when applicable, the quadratic formula provides a universal solution for any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is given by:

x = [-b ± √(b² - 4ac)] / (2a)

In our equation, 6x² + 5x - 4 = 0, we have a = 6, b = 5, and c = -4. Plugging these values into the quadratic formula, we get:

x = [-5 ± √(5² - 4 * 6 * -4)] / (2 * 6)

x = [-5 ± √(25 + 96)] / 12

x = [-5 ± √121] / 12

Since √121 = 11, we have:

x = [-5 ± 11] / 12

This gives us two possible solutions:

x₁ = (-5 + 11) / 12 = 6 / 12 = 1/2

x₂ = (-5 - 11) / 12 = -16 / 12 = -4/3

Thus, using the quadratic formula, we find the solutions to be x = 1/2 and x = -4/3, which confirms our results obtained through factoring. This demonstrates the versatility and reliability of the quadratic formula in solving quadratic equations.

H2: Verifying the Solutions

To ensure the accuracy of our solutions, it's crucial to verify them by substituting them back into the original equation. This step helps us catch any potential errors made during the solving process. Let's verify our solutions, x = -4/3 and x = 1/2, for the equation 6x² + 5x - 4 = 0.

Verification for x = -4/3:

Substitute x = -4/3 into the equation:

6(-4/3)² + 5(-4/3) - 4 = 0

6(16/9) - 20/3 - 4 = 0

32/3 - 20/3 - 12/3 = 0

(32 - 20 - 12) / 3 = 0

0 / 3 = 0

0 = 0

The equation holds true, so x = -4/3 is a valid solution.

Verification for x = 1/2:

Substitute x = 1/2 into the equation:

6(1/2)² + 5(1/2) - 4 = 0

6(1/4) + 5/2 - 4 = 0

3/2 + 5/2 - 8/2 = 0

(3 + 5 - 8) / 2 = 0

0 / 2 = 0

0 = 0

The equation holds true, so x = 1/2 is also a valid solution.

By verifying both solutions, we can confidently conclude that x = -4/3 and x = 1/2 are indeed the correct roots of the equation 6x² + 5x - 4 = 0. This verification process underscores the importance of checking our work to ensure accuracy in mathematical problem-solving.

H2: Identifying the Correct Options

Now that we have solved the equation 6x² + 5x - 4 = 0 and verified the solutions, let's identify the correct options from the given choices. The solutions we found are x = -4/3 and x = 1/2. Comparing these with the options provided, we can clearly see which ones match our solutions.

The options provided were:

A. x = -1/3

B. x = -4/3

C. x = -2

D. x = 4

E. x = 1/2

F. x = 1/3

By comparing our solutions with the options, we can identify the correct answers:

• Option B: x = -4/3 matches one of our solutions.
• Option E: x = 1/2 matches our other solution.

Therefore, the correct options are B and E. This final step of matching the calculated solutions with the provided options reinforces the importance of accuracy and attention to detail in problem-solving. It ensures that we not only solve the equation correctly but also identify the correct answers within the given context.

H2: Conclusion

In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation 6x² + 5x - 4 = 0. We explored two primary methods: factoring and the quadratic formula, both leading us to the same solutions. We also emphasized the crucial step of verifying the solutions to ensure accuracy. Through factoring, we rewrote the equation as (3x + 4)(2x - 1) = 0, yielding the solutions x = -4/3 and x = 1/2. Applying the quadratic formula, we arrived at the same solutions, further validating our results. The verification process confirmed that both x = -4/3 and x = 1/2 satisfy the original equation. Finally, we identified the correct options from the given choices, solidifying our understanding of the solution set.

Mastering the art of solving quadratic equations is a cornerstone of algebraic proficiency. This guide has not only provided the solutions to the specific equation but has also equipped you with the knowledge and skills to tackle a wide array of quadratic equations. Whether you prefer the elegance of factoring or the versatility of the quadratic formula, understanding these methods empowers you to confidently solve these equations. Remember, practice is key to mastering any mathematical concept, so continue to explore and challenge yourself with different types of quadratic equations. By doing so, you'll strengthen your algebraic foundation and enhance your problem-solving abilities.

H3: Additional Practice Problems

To further solidify your understanding of solving quadratic equations, here are some additional practice problems. Try solving these using both factoring and the quadratic formula to reinforce your skills:

1. 2x² - 5x + 2 = 0
2. 3x² + 7x - 6 = 0
3. 4x² - 9 = 0
4. x² + 6x + 9 = 0
5. 5x² - 13x + 6 = 0

Work through these problems step-by-step, and remember to verify your solutions. The more you practice, the more confident you'll become in your ability to solve quadratic equations.