Solving 3x(x+4) + 2 = 0 Using The Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions (also called roots) of any quadratic equation, which is an equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. In this article, we will walk through the process of solving the quadratic equation 3x(x+4) + 2 = 0 using this formula. We will break down each step, ensuring a clear understanding of how to apply the quadratic formula effectively.

Understanding the Quadratic Formula

Before we dive into solving the equation, let's first understand the quadratic formula itself. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

Where:

• x represents the solutions or roots of the quadratic equation.
• a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
• The symbol '±' indicates that there are two possible solutions: one using the plus sign and one using the minus sign.
• The expression b² - 4ac is called the discriminant. The discriminant tells us about the nature of the roots:
  • If b² - 4ac > 0, there are two distinct real roots.
  • If b² - 4ac = 0, there is exactly one real root (a repeated root).
  • If b² - 4ac < 0, there are no real roots (two complex roots).

The quadratic formula is derived by completing the square on the general form of the quadratic equation, ax² + bx + c = 0. It's a fundamental formula in algebra and is used extensively in various fields of mathematics, physics, engineering, and computer science. Understanding and applying this formula correctly is crucial for solving quadratic equations efficiently.

Step 1: Rewrite the Equation in Standard Form

Our given equation is 3x(x+4) + 2 = 0. The first step in applying the quadratic formula is to rewrite the equation in the standard quadratic form, which is ax² + bx + c = 0. To do this, we need to expand and simplify the equation.

First, distribute the 3x across the terms inside the parentheses:

3x(x+4) + 2 = 3x² + 12x + 2 = 0

Now, we have the equation in the standard quadratic form:

3x² + 12x + 2 = 0

Here, we can identify the coefficients:

• a = 3
• b = 12
• c = 2

Rewriting the equation in standard form is a crucial step because it allows us to easily identify the coefficients a, b, and c, which are required for the quadratic formula. Without this step, it would be difficult to correctly substitute the values into the formula.

Step 2: Identify the Coefficients a, b, and c

Now that we have the equation in the standard form, 3x² + 12x + 2 = 0, we can easily identify the coefficients a, b, and c. These coefficients are the numerical values that multiply the variables and the constant term in the quadratic equation. As we determined in the previous step:

• a is the coefficient of the x² term, which is 3.
• b is the coefficient of the x term, which is 12.
• c is the constant term, which is 2.

Correctly identifying these coefficients is crucial because they will be substituted into the quadratic formula. A mistake in identifying the coefficients will lead to an incorrect solution. Once we have these values, we can proceed to the next step, which involves plugging them into the quadratic formula.

Step 3: Apply the Quadratic Formula

With the coefficients identified (a = 3, b = 12, c = 2), we can now apply the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substitute the values of a, b, and c into the formula:

x = (-12 ± √(12² - 4 * 3 * 2)) / (2 * 3)

Now, simplify the expression step by step.

First, calculate the square of b (12²) and the product 4 * a c (4 * 3 * 2):

x = (-12 ± √(144 - 24)) / 6

Next, subtract 24 from 144:

x = (-12 ± √120) / 6

Now, we need to simplify the square root of 120. We can do this by finding the prime factorization of 120. 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5. We can rewrite √120 as √(2² × 2 × 3 × 5) = 2√(30).

x = (-12 ± 2√30) / 6

Finally, we can simplify the entire expression by dividing both terms in the numerator by the denominator (6):

x = (-12 / 6) ± (2√30 / 6)

x = -2 ± (√30 / 3)

So, we have two solutions:

x₁ = -2 + (√30 / 3)

x₂ = -2 - (√30 / 3)

Applying the quadratic formula involves careful substitution and simplification. It's essential to follow each step meticulously to avoid errors. Once the formula is applied correctly, the solutions can be found by simplifying the resulting expression. In this case, we have two distinct real solutions, as the discriminant (120) was positive.

Step 4: Simplify the Solutions

In the previous step, we found the solutions to the quadratic equation using the quadratic formula. The solutions we obtained were:

x₁ = -2 + (√30 / 3)

x₂ = -2 - (√30 / 3)

These solutions are already in their simplest form. However, it's always a good practice to double-check and ensure that there are no further simplifications possible. In this case, the square root term, √30, cannot be simplified further because 30 does not have any perfect square factors other than 1. The fraction √30 / 3 is also in its simplest form because the numerator and denominator do not share any common factors.

The solutions are exact values, and they represent the points where the quadratic equation intersects the x-axis when graphed. These exact solutions are crucial for applications where precision is required.

Step 5: Express the Exact Solutions

We have already found and simplified the exact solutions to the quadratic equation 3x(x+4) + 2 = 0. The solutions are:

x₁ = -2 + (√30 / 3)

x₂ = -2 - (√30 / 3)

To express these solutions in a comma-separated format, we simply write them as:

-2 + (√30 / 3), -2 - (√30 / 3)

This format is commonly used when providing multiple solutions to a mathematical problem. Each solution is separated by a comma, making it clear that these are two distinct roots of the equation.

In summary, solving a quadratic equation using the quadratic formula involves several steps: rewriting the equation in standard form, identifying the coefficients, applying the formula, simplifying the expression, and expressing the solutions in the requested format. By following these steps carefully, you can accurately solve any quadratic equation.

Conclusion

In this article, we have demonstrated how to solve the quadratic equation 3x(x+4) + 2 = 0 using the quadratic formula. We began by rewriting the equation in standard form, which allowed us to identify the coefficients a, b, and c. We then substituted these values into the quadratic formula and simplified the resulting expression to find the exact solutions. The solutions were found to be x₁ = -2 + (√30 / 3) and x₂ = -2 - (√30 / 3).

The quadratic formula is a versatile and reliable method for solving quadratic equations, regardless of whether the roots are real or complex. By understanding and mastering this formula, you can tackle a wide range of algebraic problems with confidence. Remember to always double-check your work and ensure that your solutions are in the simplest form. This approach not only helps in solving equations accurately but also enhances your problem-solving skills in mathematics and related fields. The ability to solve quadratic equations is a fundamental skill that is widely applicable in various areas of science, engineering, and everyday life.