Solutions Of 8x² = 6 + 22x Two Correct Answers

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Introduction

In this article, we will delve into the process of solving the quadratic equation 8x² = 6 + 22x. Quadratic equations, characterized by their second-degree polynomial form, are fundamental in mathematics and have wide-ranging applications in various fields such as physics, engineering, and economics. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. We will explore the standard methods for solving quadratic equations, including rearranging the equation into standard form, factoring, and using the quadratic formula. Our primary goal is to find the two correct solutions from the given options: x = -3, x = 6, x = 3, and x = -1/4. By the end of this guide, you will have a clear understanding of how to solve this specific equation and similar quadratic equations effectively.

Understanding Quadratic Equations

Before diving into the solution, it’s essential to understand what a quadratic equation is and its standard form. 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 coefficients a, b, and c play critical roles in determining the nature and values of the solutions, also known as roots or zeros of the equation. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. Understanding this standard form is the first step in solving any quadratic equation because many solution methods rely on this structure. For example, the quadratic formula, a universal method for finding solutions, directly uses the values of a, b, and c. Additionally, factoring techniques also depend on recognizing the relationships between these coefficients to break down the equation into simpler terms. Recognizing the standard form helps in identifying the appropriate method to solve the equation efficiently. In our case, the given equation 8x² = 6 + 22x needs to be rearranged into this standard form before we can apply any solution techniques. This rearrangement involves moving all terms to one side of the equation, setting it equal to zero, and then identifying the values of a, b, and c. This foundational understanding ensures that we approach the problem systematically and accurately, paving the way for successfully finding the solutions.

Rearranging the Equation into Standard Form

To solve the quadratic equation 8x² = 6 + 22x, the first crucial step is to rearrange it into the standard form, which is ax² + bx + c = 0. This standard form is essential because it allows us to easily identify the coefficients a, b, and c, which are necessary for applying various solution methods, such as factoring or using the quadratic formula. To rearrange the given equation, we need to move all terms to one side, leaving zero on the other side. In the equation 8x² = 6 + 22x, we begin by subtracting 22x from both sides, resulting in 8x² - 22x = 6. Next, we subtract 6 from both sides to set the equation equal to zero, giving us the standard form: 8x² - 22x - 6 = 0. Now that the equation is in the standard form, we can clearly identify the coefficients: a = 8, b = -22, and c = -6. These values are critical for the next steps in solving the equation, whether we choose to factor, complete the square, or use the quadratic formula. Rearranging the equation not only prepares it for solving but also ensures that we apply the correct methods with the correct values, which is vital for obtaining accurate solutions. This systematic approach is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems beyond quadratic equations.

Simplifying the Equation

Before applying any solution methods to the quadratic equation 8x² - 22x - 6 = 0, it is often beneficial to simplify the equation if possible. Simplifying can make the coefficients smaller and easier to work with, which can reduce the complexity of factoring or using the quadratic formula. In this case, we observe that all the coefficients (8, -22, and -6) are even numbers, which means they are all divisible by 2. Dividing each term in the equation by 2 will simplify the equation without changing its solutions. Performing this division, we get: (8x²)/2 - (22x)/2 - 6/2 = 0/2, which simplifies to 4x² - 11x - 3 = 0. Now, the simplified equation 4x² - 11x - 3 = 0 has smaller coefficients (a = 4, b = -11, and c = -3) compared to the original equation. This simplification makes the subsequent steps, such as factoring or applying the quadratic formula, more manageable. For example, when factoring, smaller numbers reduce the number of possible factor pairs to consider. When using the quadratic formula, smaller coefficients result in simpler arithmetic, decreasing the chances of making errors. Simplifying an equation before solving is a valuable technique in algebra. It not only makes the calculations easier but also demonstrates a good understanding of mathematical principles and problem-solving strategies. In the following sections, we will use this simplified equation to find the solutions, illustrating the practical benefits of this simplification step.

Method 1: Factoring the Quadratic Equation

Factoring is a method used to solve quadratic equations by expressing the quadratic expression as a product of two binomials. This method is most effective when the quadratic equation can be easily factored. To factor the simplified quadratic equation 4x² - 11x - 3 = 0, we look for two binomials that, when multiplied, give us the original quadratic expression. The general form of these binomials is (Ax + B)(Cx + D), where A, B, C, and D are constants. We need to find values for these constants such that:

(Ax + B)(Cx + D) = 4x² - 11x - 3

To do this, we can use the following steps:

  1. Identify the factors of the leading coefficient (a) and the constant term (c).
    • The leading coefficient is 4, and its factors are (1, 4) and (2, 2).
    • The constant term is -3, and its factors are (1, -3) and (-1, 3).
  2. Find a combination of factors that, when multiplied and added in the right way, give us the middle coefficient (b), which is -11.
    • We need to find a combination such that the outer and inner products of the binomials add up to -11x.
  3. Test different combinations:
    • Let's try (4x + 1)(x - 3): The outer product is -12x, and the inner product is +1x. Adding these gives -11x, which is what we need.
  4. Write the factored form of the equation:
    • Thus, the factored form of 4x² - 11x - 3 = 0 is (4x + 1)(x - 3) = 0.

Now that we have the factored form, we can use the zero-product property to find the solutions. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

  1. 4x + 1 = 0
    • Subtract 1 from both sides: 4x = -1
    • Divide by 4: x = -1/4
  2. x - 3 = 0
    • Add 3 to both sides: x = 3

So, the solutions to the quadratic equation 4x² - 11x - 3 = 0, and consequently to the original equation 8x² = 6 + 22x, are x = -1/4 and x = 3. Factoring is a powerful technique when it can be applied, as it provides a straightforward path to the solutions. However, not all quadratic equations can be easily factored, which is why it’s essential to have alternative methods like the quadratic formula at our disposal.

Method 2: Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. Unlike factoring, which can be challenging for certain equations, the quadratic formula always provides a solution, regardless of whether the equation can be factored easily. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation. To apply this formula, we first need to identify the values of a, b, and c from the simplified equation 4x² - 11x - 3 = 0. In this case, a = 4, b = -11, and c = -3. Now, we substitute these values into the quadratic formula:

x = (-(-11) ± √((-11)² - 4 * 4 * (-3))) / (2 * 4)

Let's simplify the expression step by step:

  1. Simplify the terms inside the square root:
    • (-11)² = 121
    • 4 * 4 * (-3) = -48
    • 121 - (-48) = 121 + 48 = 169
  2. Substitute these values back into the formula:
    • x = (11 ± √169) / 8
  3. Calculate the square root of 169:
    • √169 = 13
  4. Substitute the square root back into the formula:
    • x = (11 ± 13) / 8

Now we have two possible solutions, one with addition and one with subtraction:

  1. For the addition case:
    • x = (11 + 13) / 8 = 24 / 8 = 3
  2. For the subtraction case:
    • x = (11 - 13) / 8 = -2 / 8 = -1/4

Thus, the solutions to the quadratic equation 4x² - 11x - 3 = 0, and consequently to the original equation 8x² = 6 + 22x, are x = 3 and x = -1/4. The quadratic formula provides a systematic and reliable method for finding the solutions of any quadratic equation, making it an essential tool in algebra. In this case, it confirms the solutions we found using factoring, reinforcing the accuracy of our results. Whether an equation is easily factorable or not, the quadratic formula can be applied to find the roots, demonstrating its versatility and importance in solving quadratic equations.

Verifying the Solutions

After obtaining the solutions to a quadratic equation, it is crucial to verify their correctness. This step ensures that the values we found are indeed the roots of the equation and that no errors were made during the solving process. To verify the solutions x = 3 and x = -1/4 for the original equation 8x² = 6 + 22x, we substitute each value back into the equation and check if it holds true. Let's start with x = 3:

  1. Substitute x = 3 into the equation:
    • 8(3)² = 6 + 22(3)
  2. Simplify both sides:
    • 8(9) = 6 + 66
    • 72 = 72

Since the equation holds true, x = 3 is indeed a solution. Now, let's verify the second solution, x = -1/4:

  1. Substitute x = -1/4 into the equation:
    • 8(-1/4)² = 6 + 22(-1/4)
  2. Simplify both sides:
    • 8(1/16) = 6 - 22/4
    • 1/2 = 6 - 11/2
  3. Convert 6 to a fraction with a denominator of 2:
    • 6 = 12/2
  4. Continue simplifying:
    • 1/2 = 12/2 - 11/2
    • 1/2 = 1/2

Since the equation also holds true for x = -1/4, this value is also a valid solution. By verifying both solutions, we can confidently state that x = 3 and x = -1/4 are the correct roots of the given quadratic equation. This verification step is not just a formality; it is an essential part of the problem-solving process. It helps catch any potential mistakes in the algebraic manipulations or calculations. Moreover, it reinforces the understanding of what it means for a value to be a solution of an equation: when substituted, it makes the equation true. This practice is crucial for building solid mathematical skills and for ensuring accuracy in solving mathematical problems.

Conclusion

In conclusion, the solutions to the quadratic equation 8x² = 6 + 22x are x = 3 and x = -1/4. We arrived at these solutions by first rearranging the equation into the standard form 4x² - 11x - 3 = 0, simplifying it, and then applying two different methods: factoring and the quadratic formula. Factoring allowed us to express the quadratic expression as a product of two binomials, leading to the solutions directly through the zero-product property. The quadratic formula, a universally applicable method, provided the same solutions by substituting the coefficients into a specific formula. Furthermore, we verified these solutions by substituting them back into the original equation, confirming their correctness. This comprehensive approach not only solves the given problem but also illustrates the importance of understanding various methods for tackling quadratic equations. Each method has its strengths, and being proficient in both factoring and using the quadratic formula enhances problem-solving skills. Additionally, the emphasis on verifying solutions underscores the importance of accuracy and attention to detail in mathematics. Mastering these techniques is invaluable for anyone studying mathematics or related fields, as quadratic equations frequently appear in various contexts. The ability to solve them efficiently and accurately is a key skill that opens doors to more advanced mathematical concepts and applications.