Decoding Quadratic Equations Identifying The Correct Match

The quadratic formula stands as a cornerstone in algebra, providing a universal method to solve quadratic equations. These equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants, appear in various mathematical and real-world scenarios. The quadratic formula, expressed as x = (-b ± √(b² - 4ac)) / (2a), offers a direct pathway to find the roots, or solutions, of any quadratic equation. This formula is not just a mathematical tool; it's a testament to the elegance and efficiency of algebraic problem-solving. Understanding its application is crucial for anyone delving into mathematics, physics, engineering, and beyond. In this article, we will dissect a specific application of the quadratic formula and identify the corresponding quadratic equation, providing a clear understanding of how the formula works in practice.

In this segment, we delve into the specific instance of the quadratic formula presented to us. The formula provided is x = (-8 ± √(8² - 4(3)(-2))) / (2(3)). Our primary task is to decipher this expression and link it back to its original quadratic equation form. Each component of the formula holds significance and directly corresponds to the coefficients of the quadratic equation. Let’s break down each part meticulously:

• The Numerator's First Term: The term ‘-8’ in the numerator corresponds to ‘-b’ in the quadratic formula. This implies that the coefficient b in our original equation is 8. Remember, the formula includes a negative sign, so if we have -8 in the formula, it means b was originally 8.
• The Discriminant (The Term Under the Square Root): The expression under the square root, ‘8² - 4(3)(-2)’, is known as the discriminant. The discriminant is a critical part of the quadratic formula because it not only helps determine the solutions but also indicates the nature of the roots (real, distinct, or complex). Here, ‘8²’ corresponds to b², ‘4(3)(-2)’ corresponds to ‘4ac’. We already know b is 8. Thus, we can infer that a is 3 (from the ‘3’ in the expression) and c is -2 (from the ‘-2’ in the expression).
• The Denominator: The denominator ‘2(3)’ corresponds to ‘2a’ in the formula. This reaffirms our deduction that a is 3, as 2 multiplied by 3 equals 6, which is the denominator in the given expression.

By carefully examining each component, we can reverse-engineer the coefficients of the original quadratic equation. We have identified that a = 3, b = 8, and c = -2. These values are the key to reconstructing the quadratic equation that this formula solves.

Now armed with the values of a, b, and c, our next step is to reconstruct the original quadratic equation. Recall that the standard form of a quadratic equation is ax² + bx + c = 0. We've deduced that a = 3, b = 8, and c = -2. Substituting these values into the standard form, we get:

• 3x² + 8x + (-2) = 0

Simplifying this, the quadratic equation becomes:

• 3x² + 8x - 2 = 0

This equation is the original quadratic equation that the provided quadratic formula application is designed to solve. It encapsulates all the information embedded in the formula, from the coefficients to the constants. The process of reconstructing the equation from the formula highlights the interconnectedness of algebraic expressions and the importance of understanding the underlying structure of mathematical formulas. In this case, we started with the solution form and worked backward to find the problem, demonstrating a critical problem-solving skill in algebra.

Having reconstructed the original quadratic equation, 3x² + 8x - 2 = 0, we now turn our attention to the provided answer choices. Our goal is to identify which of the given equations matches our reconstructed equation. This step involves algebraic manipulation and comparison, ensuring that the chosen answer is indeed the correct one. It is a critical stage in the problem-solving process, as it solidifies the understanding of quadratic equations and their various forms.

Let's examine the provided options:

• Option A: -2x² - 8 = 10x - 3
• To compare this with our equation, we need to rearrange it into the standard form ax² + bx + c = 0. Adding 2x², subtracting 10x, and adding 3 to both sides, we get:
• 0 = 2x² + 10x + 5
• This equation does not match our reconstructed equation, 3x² + 8x - 2 = 0. Therefore, Option A is incorrect.
• Option B: 3x² - 8x - 10 = 4
• Rearranging this equation into the standard form, we subtract 4 from both sides:
• 3x² - 8x - 14 = 0
• This equation also does not match our reconstructed equation, 3x² + 8x - 2 = 0. Thus, Option B is incorrect.
• Option C: 3x² + 8x = 2
• Rearranging this equation into the standard form, we subtract 2 from both sides:
• 3x² + 8x - 2 = 0
• This equation perfectly matches our reconstructed equation. Therefore, Option C is the correct answer.

Through this meticulous analysis, we've successfully identified the correct answer choice by comparing each option with the original quadratic equation we reconstructed from the quadratic formula application. This methodical approach is crucial in mathematical problem-solving, ensuring accuracy and reinforcing understanding.

After a thorough examination of the answer choices and a meticulous comparison with our reconstructed quadratic equation, we arrive at the final verdict. The correct answer is Option C: 3x² + 8x = 2. This equation, when rearranged into the standard form, perfectly matches the equation we derived from the given application of the quadratic formula.

This process of identifying the correct equation underscores the importance of understanding the relationship between the quadratic formula and the standard form of quadratic equations. It highlights the skills required to manipulate algebraic expressions and the systematic approach needed to solve mathematical problems effectively. The journey from the formula application back to the original equation showcases the interconnectedness of mathematical concepts and the power of algebraic reasoning.

In conclusion, this exercise has taken us through a comprehensive exploration of the quadratic formula and its application. We began by dissecting a specific instance of the formula, meticulously identifying the components that correspond to the coefficients of a quadratic equation. We then reversed the process, reconstructing the original equation from the formula's application. Finally, we compared our reconstructed equation with the provided answer choices, successfully pinpointing the correct match. This step-by-step approach not only led us to the solution but also reinforced the fundamental principles underlying quadratic equations and the quadratic formula. Understanding the quadratic formula is crucial for success in algebra and beyond. This example has demonstrated how to apply the formula, interpret its components, and connect it back to the original equation. This mastery is essential for tackling a wide range of mathematical challenges and real-world problems where quadratic equations play a significant role.

By mastering these concepts, students and practitioners alike can confidently approach quadratic equations, armed with the knowledge and skills to solve them effectively. The quadratic formula, with its elegance and power, becomes a valuable tool in the mathematical arsenal, ready to be deployed whenever the need arises.