Identifying The Quadratic Equation Solved By A Specific Formula Expression

In the realm of algebra, the quadratic formula stands as a cornerstone for solving equations of the form ax² + bx + c = 0. This powerful formula provides a direct method for finding the roots, or solutions, of any quadratic equation, regardless of its complexity. The expression presented,

(-b ± √(b² - 4ac)) / 2a,

is the quintessential quadratic formula. Understanding how to dissect this formula and apply it is crucial for navigating various mathematical problems. This article aims to delve into the given expression: (-3 ± √((3)² + 4(10)(2))) / 2(10) and identify the quadratic equation it solves. We will dissect each component of the expression, relate it back to the standard quadratic formula, and then match these components with the coefficients of the quadratic equations provided in the options.

This exploration is not just about finding the correct answer; it's about deepening our comprehension of quadratic equations and the elegant formula that unlocks their solutions. By breaking down the expression and comparing it with the general form of the quadratic formula, we will develop a systematic approach to solving such problems. This method will not only help in answering the specific question at hand but also in tackling more complex quadratic equation challenges in the future. So, let's embark on this mathematical journey, unravel the mysteries of the quadratic formula, and hone our problem-solving skills.

To effectively tackle the problem at hand, a thorough understanding of the quadratic formula is essential. The quadratic formula, represented as

x = (-b ± √(b² - 4ac)) / 2a,

is the key to solving quadratic equations of the form ax² + bx + c = 0. Each term in the formula corresponds directly to the coefficients in the quadratic equation, making it a powerful tool for finding solutions.

Let's break down each component of the formula:

• a: This represents the coefficient of the x² term in the quadratic equation. It dictates the parabola's width and direction (whether it opens upwards or downwards).
• b: This is the coefficient of the x term. It influences the position of the parabola's axis of symmetry.
• c: This is the constant term in the equation. It determines the parabola's vertical position on the coordinate plane.
• (-b ± √(b² - 4ac)): This part of the formula is crucial for determining the roots of the equation. The ± sign indicates that there can be two possible solutions, one with addition and one with subtraction.
• b² - 4ac: Known as the discriminant, this component reveals the nature of the roots. If it's positive, there are two distinct real roots; if it's zero, there is exactly one real root (a repeated root); and if it's negative, there are no real roots, but two complex conjugate roots.
• √(): The square root function is applied to the discriminant, providing a value that contributes to the roots' real or complex nature.
• 2a: This term in the denominator scales the entire expression, ensuring the correct magnitude of the roots.

By grasping how each part of the formula relates to the coefficients of the quadratic equation, we can effectively reverse-engineer the equation from the solutions provided by the formula. This understanding will be pivotal in identifying the correct quadratic equation from the given options. In the following sections, we will apply this knowledge to the specific expression provided, carefully matching its components to the coefficients of the potential quadratic equations.

Now that we have a firm grasp of the quadratic formula's components, let's apply this knowledge to the given expression:

(-3 ± √((3)² + 4(10)(2))) / 2(10)

Our goal is to identify the values of a, b, and c that correspond to this expression. By carefully comparing the expression to the quadratic formula,

x = (-b ± √(b² - 4ac)) / 2a,

we can systematically deduce the coefficients of the quadratic equation.

First, let's focus on the term outside the square root in the numerator. We see -3, which corresponds to -b in the quadratic formula. This implies that b is equal to 3. Next, let's examine the denominator, 2(10), which corresponds to 2a in the formula. This clearly indicates that a is equal to 10. Now, let's turn our attention to the discriminant, the expression inside the square root:

(3)² + 4(10)(2)

In the quadratic formula, the discriminant is b² - 4ac. We already know that b is 3 and a is 10. Comparing the given expression's discriminant with the formula's, we can set up the following equation:

b² - 4ac = (3)² + 4(10)(2)

Substituting the known values of a and b, we get:

(3)² - 4(10)c = (3)² + 4(10)(2)

This simplifies to:

9 - 40c = 9 + 80

Solving for c, we subtract 9 from both sides:

-40c = 80

Finally, we divide by -40:

c = -2

Therefore, we have identified a = 10, b = 3, and c = -2. These coefficients correspond to the quadratic equation in the form ax² + bx + c = 0, which is:

10x² + 3x - 2 = 0

In the subsequent sections, we will compare this derived equation with the given options to identify the correct match, reinforcing our understanding of how to connect the quadratic formula to its corresponding equation.

Having determined the coefficients a = 10, b = 3, and c = -2 from the given expression, we now aim to identify the quadratic equation among the options that matches these coefficients. The general form of a quadratic equation is ax² + bx + c = 0. Substituting our values, we have:

10x² + 3x - 2 = 0

Now, let's examine the options provided and see which one aligns with this equation:

• Option A: 10x² + 2 = -3x

  To compare, we rearrange this equation to the standard form:

  10x² + 3x + 2 = 0

  The coefficients are a = 10, b = 3, and c = 2. This does not match our derived c = -2.

• Option B: 10x² = 3x + 2

  Rearranging to the standard form, we get:

  10x² - 3x - 2 = 0

  The coefficients are a = 10, b = -3, and c = -2. The b coefficient does not match our derived b = 3.

• Option C: 3x = 10x² - 2

  Rearranging to the standard form:

  10x² - 3x - 2 = 0

  Again, the coefficients are a = 10, b = -3, and c = -2. The b coefficient does not match.

• Option D: 2 = 3x + 10x²

  Rearranging to the standard form:

  10x² + 3x - 2 = 0

  The coefficients are a = 10, b = 3, and c = -2. This perfectly matches our derived coefficients.

Therefore, after carefully comparing the coefficients, we can confidently conclude that Option D is the correct quadratic equation that can be solved using the given expression. This exercise highlights the importance of rearranging equations into the standard form for accurate comparison and emphasizes the direct relationship between the quadratic formula and the coefficients of the quadratic equation.

After a thorough analysis, we have successfully identified Option D: 2 = 3x + 10x² as the quadratic equation that can be solved using the expression (-3 ± √((3)² + 4(10)(2))) / 2(10). Let's recap the steps that led us to this conclusion and further solidify our understanding.

First, we deconstructed the given expression, recognizing it as the result of applying the quadratic formula. By comparing the expression with the general form of the quadratic formula,

x = (-b ± √(b² - 4ac)) / 2a,

we systematically identified the values of the coefficients a, b, and c. We determined that:

• a corresponds to 10
• b corresponds to 3
• c corresponds to -2

These coefficients were derived by carefully matching the components of the given expression to the corresponding terms in the quadratic formula. For instance, the denominator 2(10) directly indicated that a = 10, and the term -3 in the numerator corresponded to -b, implying b = 3. The discriminant, the expression under the square root, allowed us to solve for c by setting up the equation

b² - 4ac = (3)² + 4(10)(2)

and solving for c.

Next, with a = 10, b = 3, and c = -2, we constructed the quadratic equation in the standard form ax² + bx + c = 0:

10x² + 3x - 2 = 0

Finally, we compared this derived equation with the provided options. To ensure accurate comparison, we rearranged each option into the standard form. It became evident that only Option D: 2 = 3x + 10x², when rearranged to 10x² + 3x - 2 = 0, perfectly matched our derived equation. The other options had discrepancies in either the b or c coefficients, making them incorrect.

This methodical approach, from dissecting the quadratic formula to matching coefficients, underscores the importance of a systematic strategy in solving mathematical problems. By understanding the underlying principles and applying them rigorously, we can confidently navigate complex challenges and arrive at the correct solution.

In conclusion, we have successfully navigated the process of identifying the quadratic equation that corresponds to a given expression derived from the quadratic formula. This journey has not only provided us with the answer to the specific question but has also reinforced our understanding of quadratic equations and the quadratic formula itself. We have seen how the quadratic formula serves as a powerful tool for solving equations of the form ax² + bx + c = 0, and how each component of the formula is intricately linked to the coefficients of the equation.

The key to solving this type of problem lies in the ability to dissect the given expression and systematically match its components to the corresponding terms in the quadratic formula. By carefully identifying the values of a, b, and c, we can reconstruct the quadratic equation and compare it with the provided options. This process requires a clear understanding of the quadratic formula, its structure, and the significance of each term.

Furthermore, this exercise highlights the importance of rearranging equations into the standard form ax² + bx + c = 0 for accurate comparison. This standard form provides a consistent framework for identifying the coefficients and ensures that we are comparing like terms with like terms.

The ability to solve quadratic equations is a fundamental skill in mathematics, with applications spanning various fields, from physics and engineering to economics and computer science. By mastering the quadratic formula and developing a systematic approach to problem-solving, we equip ourselves with a valuable tool that will serve us well in a multitude of contexts. As we continue our mathematical journey, let us remember the lessons learned here: the power of the quadratic formula, the importance of systematic analysis, and the value of a strong foundation in algebraic principles.