Identifying Quadratic Equations Solved By A Specific Quadratic Formula Expression

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In the realm of mathematics, quadratic equations hold a significant place, appearing in various contexts from simple algebraic problems to complex scientific models. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form $ax^2 + bx + c = 0$, where a, b, and c are constants, and x represents the unknown variable. Solving quadratic equations involves finding the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. Among the several methods available for solving quadratic equations, the quadratic formula stands out as a powerful and universally applicable technique. This formula provides a direct and efficient way to determine the solutions, regardless of the specific values of the coefficients a, b, and c. Understanding the quadratic formula and its application is crucial for anyone delving into algebra and beyond.

The quadratic formula is derived from the process of completing the square, a method that transforms the quadratic equation into a perfect square trinomial. The formula itself is expressed as: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula elegantly encapsulates the relationship between the coefficients of the quadratic equation and its solutions. The expression under the square root, $b^2 - 4ac$, is known as the discriminant, and it plays a pivotal role in determining the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (or two equal real roots), and a negative discriminant indicates two complex roots.

In this article, we will explore how to identify the quadratic equation that corresponds to a given solution expression derived from the quadratic formula. Specifically, we will focus on the expression $\frac{-3 \pm \sqrt{(3)^2 + 4(10)(2)}}{2(10)}$ and determine which quadratic equation among the provided options can be solved using this expression. This exercise will not only reinforce our understanding of the quadratic formula but also enhance our ability to connect the solutions back to the original equation. We will dissect the given expression, identify the values of a, b, and c, and then reconstruct the corresponding quadratic equation. This process will underscore the power and versatility of the quadratic formula as a tool for solving a wide range of quadratic equations.

Dissecting the Solution Expression and Reconstructing the Quadratic Equation

To decipher which quadratic equation can be solved using the provided expression, we need to meticulously examine the solution expression and map its components back to the quadratic formula. The given expression is $\frac{-3 \pm \sqrt{(3)^2 + 4(10)(2)}}{2(10)}$. By comparing this expression to the general form of the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we can extract the values of a, b, and c. The denominator, 2(10), directly corresponds to 2a, which implies that a = 10. The term -3 in the numerator corresponds to -b, indicating that b = 3. Finally, the expression under the square root, (3)^2 + 4(10)(2), corresponds to $b^2 - 4ac$. Since we already know that a = 10 and b = 3, we can deduce the value of c. The expression can be rewritten as 3^2 - 4(10)(-2), which means that c = -2.

Now that we have identified the coefficients a = 10, b = 3, and c = -2, we can reconstruct the quadratic equation in the standard form $ax^2 + bx + c = 0$. Substituting the values, we get $10x^2 + 3x - 2 = 0$. This equation represents the original quadratic equation that would yield the given solution expression when solved using the quadratic formula. Therefore, to identify the correct option, we need to rearrange the given equations and see which one matches our reconstructed equation. This process involves algebraic manipulation and a careful comparison of terms.

This step of reconstructing the quadratic equation from its solution expression is a critical skill in algebra. It demonstrates the inverse relationship between the coefficients of the equation and its roots. Understanding this relationship allows us to not only solve quadratic equations but also to manipulate and analyze them in various ways. For instance, we can use this skill to create quadratic equations with specific solutions or to analyze the behavior of quadratic functions. The ability to dissect a solution expression and reconstruct the original equation is a testament to a deep understanding of quadratic equations and their properties. This understanding is crucial for success in higher-level mathematics and related fields.

Analyzing the Options and Identifying the Correct Equation

Having reconstructed the quadratic equation as $10x^2 + 3x - 2 = 0$, our next step is to compare this equation with the given options and identify the correct match. This involves carefully examining each option and rearranging it, if necessary, to see if it aligns with our reconstructed equation. The process requires a keen eye for detail and a solid understanding of algebraic manipulation. Each option presents a slightly different arrangement of terms, and it is essential to ensure that the equation is in the standard form $ax^2 + bx + c = 0$ for accurate comparison.

Let's analyze the options one by one:

  • Option A: $10x^2 = 3x + 2$. To compare this with our equation, we need to move all terms to one side. Subtracting $3x$ and 2 from both sides, we get $10x^2 - 3x - 2 = 0$. This equation does not match our reconstructed equation, $10x^2 + 3x - 2 = 0$, as the sign of the bx term is different.
  • Option B: $2 = 3x + 10x^2$. Rearranging the terms to match the standard form, we get $10x^2 + 3x - 2 = 0$. This equation perfectly matches our reconstructed equation. Therefore, this option is a potential solution.
  • Option C: $3x = 10x^2 - 2$. Rearranging the terms, we get $10x^2 - 3x - 2 = 0$. This equation does not match our reconstructed equation due to the sign of the bx term.
  • Option D: $10x^2 + 2 = -3x$. Rearranging the terms, we get $10x^2 + 3x + 2 = 0$. This equation does not match our reconstructed equation as the sign of the constant term c is different.

After careful analysis, we can conclude that Option B, $2 = 3x + 10x^2$, is the correct equation that can be solved using the given expression. This process highlights the importance of algebraic manipulation and careful comparison when working with quadratic equations. It also reinforces the understanding that different arrangements of the same equation can be misleading if not properly analyzed. This skill is not only crucial for solving quadratic equations but also for tackling more complex algebraic problems.

Conclusion Understanding the Power of the Quadratic Formula

In this exploration, we successfully identified the quadratic equation that corresponds to the solution expression $\frac{-3 \pm \sqrt{(3)^2 + 4(10)(2)}}{2(10)}$. By dissecting the expression, extracting the coefficients a, b, and c, and reconstructing the quadratic equation, we demonstrated the power and versatility of the quadratic formula. This exercise underscores the importance of understanding the relationship between the coefficients of a quadratic equation and its solutions. We carefully analyzed each option and, through algebraic manipulation and comparison, determined that Option B, $2 = 3x + 10x^2$, is the correct equation.

The quadratic formula is a cornerstone of algebra and a fundamental tool for solving a wide range of problems. Its ability to provide solutions for any quadratic equation, regardless of the complexity of its coefficients, makes it an indispensable asset in mathematics and related fields. The discriminant, the expression under the square root in the quadratic formula, provides valuable information about the nature of the roots, allowing us to predict whether the equation will have two distinct real roots, one real root, or two complex roots. This understanding is crucial for analyzing the behavior of quadratic functions and their applications in various contexts.

The ability to connect a solution expression back to its original equation is a testament to a deep understanding of quadratic equations. It highlights the inverse relationship between the coefficients and the roots, allowing us to not only solve equations but also to manipulate and analyze them effectively. This skill is essential for success in higher-level mathematics, including calculus, differential equations, and linear algebra. Furthermore, quadratic equations and the quadratic formula have numerous applications in real-world scenarios, such as physics, engineering, economics, and computer science. From modeling projectile motion to optimizing financial investments, the principles of quadratic equations are widely applicable.

In conclusion, mastering the quadratic formula and its applications is a valuable investment in one's mathematical journey. It provides a powerful tool for solving quadratic equations, understanding their properties, and applying them to a variety of real-world problems. By carefully analyzing the solution expression and comparing it to the general form of the quadratic formula, we were able to successfully identify the correct equation. This exercise reinforces the importance of precision, algebraic manipulation, and a deep understanding of mathematical concepts.

  • The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$.
  • By comparing a solution expression to the quadratic formula, we can extract the coefficients a, b, and c and reconstruct the original quadratic equation.
  • Careful algebraic manipulation and comparison are essential for identifying the correct equation from a set of options.
  • Understanding the quadratic formula and its applications is crucial for success in higher-level mathematics and related fields.