Solving 8x² = 6 + 22x A Step By Step Guide

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Introduction

In this article, we will delve into the process of solving the quadratic equation 8x² = 6 + 22x. Quadratic equations, which are polynomial equations of the second degree, play a pivotal role in various fields, including mathematics, physics, engineering, and economics. Mastering the techniques to solve these equations is crucial for anyone seeking a deeper understanding of these disciplines. We will explore several methods to find the solutions, also known as roots, of the given equation. Furthermore, we will meticulously check each potential solution to ensure accuracy and reinforce the understanding of the solution-verification process. This article aims to provide a clear, step-by-step guide suitable for both students and professionals seeking to enhance their problem-solving skills in mathematics.

Transforming the Equation into Standard Form

Before we embark on solving the equation 8x² = 6 + 22x, it is essential to transform it into the standard quadratic form, which is ax² + bx + c = 0. This form allows us to readily apply various solution methods, such as factoring, completing the square, or using the quadratic formula. The standard form provides a structured approach to identifying the coefficients and constants, which are vital for these methods. By rearranging the terms, we can rewrite the given equation and set the stage for solving it effectively. This initial step is fundamental to ensuring that we approach the problem in a systematic and organized manner, paving the way for a successful solution.

To transform the equation 8x² = 6 + 22x into the standard form, we need to move all terms to one side, leaving zero on the other side. This involves subtracting 22x and 6 from both sides of the equation. The steps are as follows:

  1. Start with the original equation: 8x² = 6 + 22x.
  2. Subtract 22x from both sides: 8x² - 22x = 6.
  3. Subtract 6 from both sides: 8x² - 22x - 6 = 0.

Now, the equation is in the standard quadratic form ax² + bx + c = 0, where a = 8, b = -22, and c = -6. This form allows us to proceed with various methods to find the solutions for x.

Simplifying the Equation

Before applying any solution methods, simplifying the quadratic equation can significantly ease the calculations and reduce the chances of errors. In the equation 8x² - 22x - 6 = 0, we notice that all the coefficients (8, -22, and -6) are divisible by 2. Dividing the entire equation by their greatest common divisor simplifies the equation without changing its solutions. This process makes the numbers smaller and more manageable, thereby making the subsequent steps less cumbersome. Simplifying the equation is a practical step that can save time and effort while ensuring accuracy in the solution process.

To simplify the equation 8x² - 22x - 6 = 0, we divide all terms by the greatest common divisor of the coefficients, which is 2:

(8x² - 22x - 6) / 2 = 0 / 2

This simplifies to:

4x² - 11x - 3 = 0

Now, the simplified equation 4x² - 11x - 3 = 0 is easier to work with. The coefficients are smaller, making the factoring process, if applicable, more straightforward, and the quadratic formula less prone to computational errors. This step of simplification is a valuable technique in solving quadratic equations efficiently.

Solving by Factoring

Factoring is a powerful method for solving quadratic equations, particularly when the equation can be expressed as a product of two binomials. This method relies on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Factoring involves breaking down the quadratic expression into its constituent binomial factors, setting each factor equal to zero, and solving for x. This approach is often quicker and more intuitive than other methods, especially when the factors are readily apparent. However, not all quadratic equations can be easily factored, making it essential to understand alternative methods as well. Factoring provides a direct and elegant way to find the solutions when applicable, making it a fundamental skill in algebra.

To solve the simplified quadratic equation 4x² - 11x - 3 = 0 by factoring, we need to find two binomials that multiply to give the quadratic expression. We look for two numbers that multiply to the product of the leading coefficient (4) and the constant term (-3), which is -12, and add up to the middle coefficient (-11). These numbers are -12 and 1.

We can rewrite the middle term using these numbers:

4x² - 12x + x - 3 = 0

Now, we factor by grouping:

  • Group the first two terms and the last two terms: (4x² - 12x) + (x - 3) = 0
  • Factor out the greatest common factor from each group: 4x(x - 3) + 1(x - 3) = 0
  • Notice that (x - 3) is a common factor, so we factor it out: (4x + 1)(x - 3) = 0

Now, we apply the zero-product property by setting each factor equal to zero:

  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

Thus, the solutions to the quadratic equation are x = -1/4 and x = 3. Factoring provides a straightforward method for solving quadratic equations when the factors are easily identifiable.

Verifying the Solutions

After finding potential solutions to a quadratic equation, it is imperative to verify them by substituting each value back into the original equation. This step ensures that the solutions satisfy the equation and that no errors were made during the solving process. Verification is a crucial part of problem-solving in mathematics, as it provides a check for accuracy and completeness. By plugging the solutions back into the equation, we can confirm that the left-hand side equals the right-hand side, thus validating the solutions. This process not only confirms the correctness of the answers but also reinforces the understanding of how solutions relate to the equation itself.

To verify the solutions x = -1/4 and x = 3 for the original equation 8x² = 6 + 22x, we substitute each value into the equation and check if it holds true.

  1. For x = -1/4:

    • Substitute x = -1/4 into the equation: 8(-1/4)² = 6 + 22(-1/4)
    • Simplify the left-hand side: 8(1/16) = 1/2
    • Simplify the right-hand side: 6 - 22/4 = 6 - 11/2 = (12 - 11)/2 = 1/2
    • Since the left-hand side equals the right-hand side (1/2 = 1/2), x = -1/4 is a valid solution.

  2. For x = 3:

    • Substitute x = 3 into the equation: 8(3)² = 6 + 22(3)
    • Simplify the left-hand side: 8(9) = 72
    • Simplify the right-hand side: 6 + 66 = 72
    • Since the left-hand side equals the right-hand side (72 = 72), x = 3 is a valid solution.

Both x = -1/4 and x = 3 satisfy the original equation, confirming that they are indeed the correct solutions. This verification step underscores the importance of checking solutions to ensure accuracy in mathematical problem-solving.

Conclusion

In summary, we have successfully solved the quadratic equation 8x² = 6 + 22x by transforming it into standard form, simplifying it, and then factoring. The solutions obtained are x = -1/4 and x = 3. We also verified these solutions by substituting them back into the original equation, confirming their validity. This process highlights the importance of each step in solving quadratic equations, from rearranging terms to verifying solutions. Mastering these techniques is essential for anyone working with mathematical problems in various fields. The ability to solve quadratic equations accurately and efficiently is a valuable skill that enhances problem-solving capabilities and fosters a deeper understanding of mathematical concepts.