Quadratic Equation Solver: Find The Matching Equation!

Hey guys! Let's dive into the fascinating world of quadratic equations and explore how to identify the equation that matches a given solution expression. This is a common type of problem in algebra, and mastering it will definitely boost your math skills. We're going to break down the process step-by-step, so you can confidently tackle these problems. So, let’s get started and unravel the mystery behind this question!

Understanding the Quadratic Formula

Before we jump into the specific question, let's refresh our understanding of the quadratic formula. This formula is the key to solving quadratic equations of the form ax² + bx + c = 0. The quadratic formula is expressed as:

x = (-b ± √(b² - 4ac)) / (2a)

This formula gives us the solutions (also called roots) for x in the quadratic equation. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant tells us about the nature of the roots: whether they are real or complex, and whether they are distinct or repeated. Understanding the quadratic formula is crucial because the given expression in the question is actually a direct application of this formula.

To truly grasp the quadratic formula, it's essential to understand each component and how they relate to the coefficients of the quadratic equation. The coefficient a is the number in front of the x² term, b is the number in front of the x term, and c is the constant term. The ± symbol indicates that there are typically two solutions: one where you add the square root part and one where you subtract it. This formula is not just a random set of symbols; it's a powerful tool derived from the method of completing the square, which provides a systematic way to solve any quadratic equation. Familiarizing yourself with the derivation can provide a deeper understanding of why the formula works and can help you remember it more effectively. Practice applying the formula to various quadratic equations to build your confidence and proficiency. This will make recognizing the correct equation from a given solution much easier.

Deconstructing the Given Expression

The expression we are given is:

(-3 ± √((3)² + 4(10)(2))) / (2(10)).

Our mission is to match this expression with the correct quadratic equation. To do this, we need to carefully dissect the expression and identify the values that correspond to a, b, and c in the quadratic formula. By comparing the given expression with the general quadratic formula, we can reverse-engineer the coefficients of the original quadratic equation. This involves recognizing patterns and understanding how each part of the expression relates to the quadratic formula. For instance, the number outside the square root and the coefficient in the denominator are clues to the values of b and a, respectively. The expression inside the square root provides information about the discriminant, which further helps in identifying the correct equation. This process of deconstruction is like solving a puzzle, where each piece of the expression fits into the corresponding part of the quadratic formula.

Let's break it down piece by piece:

• The numerator has "-3 ± √(...)". This suggests that -b = -3, which means b = 3. The "±" part indicates that we're dealing with two possible solutions, a characteristic feature of quadratic equations.
• The denominator is "2(10)", which corresponds to 2a. This tells us that a = 10. The denominator is crucial because it directly relates to the leading coefficient of the quadratic equation, making it a key element in our identification process.
• Inside the square root, we have "(3)² + 4(10)(2)". This corresponds to b² - 4ac. We already know b = 3 and a = 10, so this part confirms our findings and helps us determine the value of c. The expression under the square root is the discriminant, which not only helps in finding c but also gives us insight into the nature of the roots of the equation. A positive discriminant indicates real and distinct roots, while a zero discriminant indicates real and equal roots, and a negative discriminant indicates complex roots.

By carefully examining each part, we can start to piece together the original quadratic equation. This methodical approach ensures accuracy and helps in avoiding common mistakes. Understanding the significance of each component of the expression is key to successfully solving this type of problem. It's like being a detective, where each clue leads you closer to the final answer. Practice with different examples will further hone your skills in deconstructing these expressions and identifying the corresponding quadratic equations.

Reconstructing the Quadratic Equation

Now that we've identified a = 10, b = 3, and can deduce c from the square root part, let's reconstruct the quadratic equation. The square root part is √(3² + 4(10)(2)), which corresponds to √(b² - 4ac). So, we have:

• b² - 4ac = 3² + 4(10)(2)
• 3² - 4(10)c = 3² + 4(10)(2)
• -4(10)c = 4(10)(2)
• c = -2

We've now successfully identified all the coefficients: a = 10, b = 3, and c = -2. With these coefficients, we can construct the quadratic equation in the standard form ax² + bx + c = 0. Plugging in the values, we get:

10x² + 3x - 2 = 0

This process of reconstructing the equation involves putting the pieces back together like a puzzle. It demonstrates the reverse application of the quadratic formula, where we move from the solution to the original equation. This skill is valuable not only for solving this type of problem but also for understanding the relationship between the coefficients and the roots of a quadratic equation. It reinforces the idea that the quadratic formula is a central tool in algebra, connecting the solutions directly to the equation's structure. Practice in reconstructing equations from their solutions will deepen your understanding and improve your problem-solving abilities in quadratic equations and related topics.

Matching with the Options

Now, we need to match our reconstructed equation, 10x² + 3x - 2 = 0, with the given options. This step involves careful comparison and rearrangement of the given options to see which one exactly matches our equation. It's like finding the right key for a lock, where the key is our reconstructed equation and the lock is one of the options provided. The key is to ensure that the equation is in the standard form or a simple rearrangement of it. This might involve adding or subtracting terms from both sides of the equation to match it with the standard form ax² + bx + c = 0. Attention to detail is crucial here, as a small error in rearranging the terms can lead to selecting the wrong answer.

Let's look at the options:

• A. 10x² = 3x + 2 can be rewritten as 10x² - 3x - 2 = 0
• B. 2 = 3x + 10x² can be rewritten as 10x² + 3x - 2 = 0
• C. 3x = 10x² - 2 can be rewritten as 10x² - 3x - 2 = 0
• D. 10x² + 2 = -3x can be rewritten as 10x² + 3x + 2 = 0

By comparing these rearranged equations with our reconstructed equation, 10x² + 3x - 2 = 0, we can clearly see that option B, 2 = 3x + 10x², matches perfectly.

This final step is a critical check to ensure that we have not only arrived at the correct equation but also that we have correctly identified it among the given options. It emphasizes the importance of accuracy and attention to detail in problem-solving. The ability to rearrange equations and compare them effectively is a valuable skill in algebra and other areas of mathematics. It's like proofreading your work to catch any errors and ensure that your answer is correct. By practicing this matching process, you can build confidence in your ability to solve quadratic equation problems accurately and efficiently.

Conclusion

Therefore, the correct answer is B. 2 = 3x + 10x². We successfully identified the quadratic equation by deconstructing the given expression, reconstructing the equation, and then matching it with the provided options. Remember, the key to these problems is understanding the quadratic formula and how to manipulate equations. Keep practicing, and you'll become a quadratic equation pro in no time! This process of solving the problem not only gives us the answer but also enhances our understanding of quadratic equations and their properties. It's like learning a new language, where each problem solved adds to our vocabulary and grammar, making us more fluent in the language of mathematics. So, keep exploring, keep practicing, and keep growing your mathematical skills!