Solving 3x² + 8x = 12 A Step-by-Step Guide

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Quadratic equations are a fundamental topic in algebra, appearing in various mathematical and real-world applications. This article aims to provide a comprehensive guide to solving the specific quadratic equation 3x² + 8x = 12. We will explore the different methods available, with a particular focus on the quadratic formula, and provide a step-by-step solution to this equation. Understanding how to solve quadratic equations is a crucial skill for students and anyone working with mathematical models. Whether you're preparing for an exam or tackling a practical problem, mastering these techniques will prove invaluable.

Understanding Quadratic Equations

Before diving into the solution, let's establish a solid understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to 0 (if 'a' were 0, the equation would become linear). The solutions to a quadratic equation are also known as its roots or zeros. These are the values of 'x' that satisfy the equation, making the expression equal to zero. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots depends on the discriminant, which we'll discuss later.

There are several methods for solving quadratic equations, each with its own 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 efficient for simpler equations but can be challenging for more complex ones.
  2. Completing the Square: This method transforms the equation into a perfect square trinomial, allowing for a straightforward solution by taking the square root. It's a powerful technique but can be more involved than other methods.
  3. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of its complexity. It's a reliable and widely used method.

In the case of our equation, 3x² + 8x = 12, we will primarily use the quadratic formula to find the solutions. However, it's beneficial to understand the other methods as well, as they can be more efficient in certain situations. Recognizing the structure of quadratic equations and the various solution methods is the first step towards confidently tackling these problems.

Preparing the Equation for Solution

Before we can apply the quadratic formula or any other method, we need to prepare the equation 3x² + 8x = 12 into the standard quadratic form, which is ax² + bx + c = 0. This involves rearranging the terms so that all terms are on one side of the equation and the other side is zero. This step is crucial because the coefficients 'a', 'b', and 'c' are used directly in the quadratic formula. Incorrectly identifying these coefficients will lead to an incorrect solution.

To transform the equation 3x² + 8x = 12 into standard form, we need to subtract 12 from both sides of the equation. This will move the constant term to the left side and leave zero on the right side. The process is as follows:

  • Start with the given equation: 3x² + 8x = 12
  • Subtract 12 from both sides: 3x² + 8x - 12 = 12 - 12
  • Simplify: 3x² + 8x - 12 = 0

Now we have the equation in the standard form ax² + bx + c = 0. We can clearly identify the coefficients:

  • a = 3 (the coefficient of x²)
  • b = 8 (the coefficient of x)
  • c = -12 (the constant term)

Having correctly identified the coefficients is a critical step. These values will be plugged into the quadratic formula to calculate the solutions for 'x'. A simple mistake in this step can invalidate the entire solution process. Take your time and double-check your work to ensure you have the correct values for 'a', 'b', and 'c'. With the equation in standard form and the coefficients identified, we are now ready to apply the quadratic formula and find the solutions for 'x'. This preparation is essential for a successful and accurate solution.

Applying the Quadratic Formula

Now that we have the quadratic equation 3x² + 8x - 12 = 0 in standard form and have identified the coefficients a = 3, b = 8, and c = -12, we can apply the quadratic formula to find the solutions for 'x'. The quadratic formula is a powerful tool that provides a direct method for solving any quadratic equation, regardless of whether it can be factored easily. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

The quadratic formula essentially encodes the process of completing the square, providing a shortcut to finding the roots of the equation. It's a fundamental formula in algebra and should be memorized for efficient problem-solving. To apply the formula, we simply substitute the values of 'a', 'b', and 'c' that we identified earlier. Let's break down the process step-by-step:

  1. Substitute the values: Replace 'a', 'b', and 'c' in the formula with their respective values:

    x = (-8 ± √(8² - 4 * 3 * -12)) / (2 * 3)

  2. Simplify the expression under the square root: This part of the formula, b² - 4ac, is known as the discriminant. The discriminant determines the nature of the roots (real, distinct, repeated, or complex). Let's calculate it:

    8² - 4 * 3 * -12 = 64 + 144 = 208

  3. Continue simplifying: Now we substitute the discriminant back into the formula:

    x = (-8 ± √208) / 6

  4. Simplify the square root: We can simplify √208 by finding its prime factorization. 208 can be factored as 16 * 13, so √208 = √(16 * 13) = 4√13. Substituting this back into the equation:

    x = (-8 ± 4√13) / 6

  5. Reduce the fraction: We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

    x = (-4 ± 2√13) / 3

Now we have two solutions for 'x', one with the plus sign and one with the minus sign. These are the roots of the quadratic equation.

Finding the Solutions

After applying the quadratic formula to the equation 3x² + 8x - 12 = 0, we arrived at the simplified form:

x = (-4 ± 2√13) / 3

This expression represents two distinct solutions, one obtained by using the plus sign and the other by using the minus sign. These solutions are:

  1. Solution 1 (using the plus sign):

    x₁ = (-4 + 2√13) / 3

    This is one of the roots of the quadratic equation. It represents the point where the parabola represented by the equation intersects the x-axis.

  2. Solution 2 (using the minus sign):

    x₂ = (-4 - 2√13) / 3

    This is the second root of the quadratic equation, representing the other point where the parabola intersects the x-axis.

These solutions are in exact form, meaning they are expressed using the square root of 13. While this form is mathematically precise, it's often helpful to obtain approximate decimal values for practical applications. To do this, we can use a calculator to approximate the value of √13 (which is approximately 3.60555) and then perform the calculations. Let's find the approximate decimal values for our solutions:

  1. Approximate value of x₁:

    x₁ ≈ (-4 + 2 * 3.60555) / 3 ≈ (-4 + 7.2111) / 3 ≈ 3.2111 / 3 ≈ 1.0704

    So, the approximate value of the first solution is approximately 1.0704.

  2. Approximate value of x₂:

    x₂ ≈ (-4 - 2 * 3.60555) / 3 ≈ (-4 - 7.2111) / 3 ≈ -11.2111 / 3 ≈ -3.7370

    So, the approximate value of the second solution is approximately -3.7370.

Therefore, the solutions to the quadratic equation 3x² + 8x = 12 are approximately x₁ ≈ 1.0704 and x₂ ≈ -3.7370. These values represent the x-intercepts of the parabola defined by the equation. Finding both the exact form and the approximate decimal values of the solutions provides a complete understanding of the roots of the equation.

Verifying the Solutions

Once we have found the solutions to a quadratic equation, it is essential to verify the solutions to ensure they are correct. This step helps to catch any errors made during the solving process, such as mistakes in applying the quadratic formula or simplifying the expressions. Verification involves substituting each solution back into the original equation and checking if the equation holds true. If the equation holds true for a particular solution, then that solution is correct. If it does not, then there was likely an error in the solving process, and we need to revisit the steps to find the mistake.

For the equation 3x² + 8x = 12, we found two solutions:

  • x₁ = (-4 + 2√13) / 3 ≈ 1.0704
  • x₂ = (-4 - 2√13) / 3 ≈ -3.7370

Let's verify each solution by substituting it back into the original equation:

1. Verifying x₁ = (-4 + 2√13) / 3:

Substitute x₁ into the equation 3x² + 8x = 12:

3 * [((-4 + 2√13) / 3)²] + 8 * [(-4 + 2√13) / 3] = 12

This looks complex, but let's simplify step-by-step. We can also use the approximate value for a quicker check:

Using the approximate value x₁ ≈ 1.0704:

3 * (1.0704)² + 8 * 1.0704 ≈ 3 * 1.1458 + 8.5632 ≈ 3.4374 + 8.5632 ≈ 12.0006

The result is very close to 12, which confirms that x₁ is likely a correct solution. For a more rigorous verification, we would work with the exact form, but this approximate check gives us confidence.

2. Verifying x₂ = (-4 - 2√13) / 3:

Substitute x₂ into the equation 3x² + 8x = 12:

3 * [((-4 - 2√13) / 3)²] + 8 * [(-4 - 2√13) / 3] = 12

Again, let's use the approximate value x₂ ≈ -3.7370 for a quicker check:

3 * (-3.7370)² + 8 * (-3.7370) ≈ 3 * 13.9652 - 29.896 ≈ 41.8956 - 29.896 ≈ 11.9996

The result is again very close to 12, confirming that x₂ is also likely a correct solution. As with x₁, a more rigorous verification would involve working with the exact form, but the approximate check provides strong evidence.

Since both solutions, x₁ and x₂, when substituted back into the original equation, yield results very close to 12, we can confidently conclude that these are the correct solutions to the quadratic equation 3x² + 8x = 12. Verification is a crucial step in the problem-solving process, providing assurance in the accuracy of our results.

Conclusion

In this article, we've thoroughly explored the process of solving the quadratic equation 3x² + 8x = 12. We began by understanding the general form of quadratic equations and the various methods available for solving them. We then focused on using the quadratic formula, a powerful and versatile tool that can solve any quadratic equation. By carefully identifying the coefficients, substituting them into the formula, and simplifying the resulting expression, we found the two solutions to the equation. These solutions, both in exact form and approximate decimal form, represent the roots of the equation.

Furthermore, we emphasized the importance of verifying the solutions. By substituting the solutions back into the original equation, we confirmed their correctness and ensured that no errors were made during the solving process. This step is crucial for building confidence in the results and for ensuring accuracy in mathematical problem-solving.

The quadratic formula is a cornerstone of algebra, and mastering its application is essential for anyone working with mathematical models. The ability to solve quadratic equations opens doors to a wide range of applications in fields such as physics, engineering, economics, and computer science. By understanding the concepts and techniques presented in this article, you can confidently tackle quadratic equations and apply them to solve real-world problems.

Remember, practice is key to mastering any mathematical skill. Work through various examples, and don't hesitate to revisit the steps and concepts discussed in this article. With consistent effort, you'll develop a strong understanding of quadratic equations and their solutions, empowering you to succeed in your mathematical endeavors.