Solve This Equation 2s S 12 132

Solving the Equation: 2s² + 12s = 132

This article delves into the process of solving the quadratic equation 2s² + 12s = 132. We will explore multiple methods, emphasizing the importance of understanding the underlying mathematical principles. We'll cover step-by-step solutions, discuss common mistakes to avoid, and finally, offer additional practice problems to solidify your understanding. This comprehensive guide aims to equip you with the skills to tackle similar quadratic equations confidently.

Understanding Quadratic Equations

Before we dive into the solution, let's briefly review what a quadratic equation is. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're solving for. In our case, the equation 2s² + 12s = 132 is a quadratic equation, although it's not yet in the standard form. Notice that 's' is used instead of 'x' as the variable; this is simply a change of notation and doesn't affect the solving process.

Method 1: Rearranging to Standard Form and Factoring

The first step is to rearrange our equation into the standard form ax² + bx + c = 0. To achieve this, we subtract 132 from both sides of the equation:

2s² + 12s - 132 = 0

Now, we can attempt to solve this equation through factoring. Factoring involves expressing the quadratic expression as a product of two linear expressions. This method is particularly useful when the factors are easily identifiable. Let's proceed with factoring:

First, we can observe that all the coefficients (2, 12, and -132) are divisible by 2. We can simplify the equation by dividing both sides by 2:

s² + 6s - 66 = 0

Now we look for two numbers that add up to 6 (the coefficient of 's') and multiply to -66 (the constant term). After some trial and error, we find that these numbers are 12 and -6. Therefore, we can factor the equation as follows:

(s + 12)(s - 6) = 0

This equation is satisfied if either (s + 12) = 0 or (s - 6) = 0. Solving these two linear equations gives us the solutions:

• s = -12
• s = 6

Method 2: Using the Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation. It's a reliable method, especially when factoring proves difficult or impossible. The quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

Where 'a', 'b', and 'c' are the coefficients from the standard form of the quadratic equation (ax² + bx + c = 0).

In our equation, s² + 6s - 66 = 0, we have a = 1, b = 6, and c = -66. Substituting these values into the quadratic formula, we get:

s = (-6 ± √(6² - 4 * 1 * -66)) / (2 * 1)

s = (-6 ± √(36 + 264)) / 2

s = (-6 ± √300) / 2

s = (-6 ± 10√3) / 2

s = -3 ± 5√3

Therefore, the solutions are:

• s = -3 + 5√3
• s = -3 - 5√3

Note that these solutions are irrational numbers, meaning they cannot be expressed as a simple fraction. This highlights the power of the quadratic formula, as it can handle both rational and irrational solutions.

Comparing the Methods and Choosing the Best Approach

Both factoring and the quadratic formula are valid methods for solving quadratic equations. Factoring is generally quicker and easier if the equation can be easily factored. However, many quadratic equations are not easily factored, making the quadratic formula a more versatile and reliable option. The choice of method depends on the specific equation and your comfort level with each technique. For the equation 2s² + 12s = 132, factoring after simplification proves to be efficient, while the quadratic formula offers a more general and powerful approach, capable of handling more complex equations.

Common Mistakes to Avoid

Several common errors can arise when solving quadratic equations:

• Incorrect Simplification: Always ensure that you correctly simplify the equation before applying any solution method. Errors in simplifying can lead to inaccurate solutions.
• Errors in Factoring: When factoring, carefully check your work to ensure that the product of your factors correctly expands back to the original quadratic expression.
• Incorrect Application of the Quadratic Formula: Pay close attention to the signs and order of operations when substituting values into the quadratic formula. A simple sign error can drastically change the result.
• Forgetting to Check Solutions: It's always a good practice to substitute your solutions back into the original equation to verify that they are indeed correct.

Practice Problems

To solidify your understanding, try solving these quadratic equations using both factoring (if possible) and the quadratic formula:

1. x² + 5x + 6 = 0
2. 2y² - 7y + 3 = 0
3. 3z² - 12z + 12 = 0
4. t² - 4t - 12 = 0
5. 4m² + 12m + 9 = 0

Conclusion

Solving quadratic equations like 2s² + 12s = 132 is a fundamental skill in algebra. Understanding the different methods, such as factoring and the quadratic formula, and practicing regularly will build your confidence and proficiency. Remember to always check your work and avoid the common mistakes highlighted in this article. With consistent practice, you'll become adept at solving a wide range of quadratic equations. This article provides a solid foundation for further exploration of more advanced mathematical concepts. Continue to practice and refine your skills to master this important area of mathematics.