Decoding Haley's Quadratic Equation Finding The Standard Form

by ADMIN 62 views

In this article, we will dissect a problem where Haley has used the quadratic formula after substituting values for a, b, and c. Our goal is to reverse-engineer her steps and identify the original quadratic function in standard form. This requires a deep understanding of the quadratic formula and how it relates to the coefficients of a quadratic equation. This article aims to provide a step-by-step explanation, ensuring clarity and comprehension for all readers.

Understanding the Problem

Haley has applied the quadratic formula, and we are given her work:

x=−(−10)±(−10)2−4(7)(−2)2(7)x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(7)(-2)}}{2(7)}

Our mission is to determine the quadratic function in standard form, which is represented as:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

We need to find the values of the coefficients a, b, and c.

The Quadratic Formula: A Quick Review

The quadratic formula is a powerful tool for finding the solutions (or roots) of a quadratic equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. The formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula provides the values of x where the quadratic equation equals zero. The expression inside the square root, b2−4acb^2 - 4ac, is called the discriminant, and it provides valuable information about the nature of the roots.

Deconstructing Haley's Work: Identifying a, b, and c

To find the values of a, b, and c in Haley's equation, we need to carefully compare her work with the general quadratic formula. Let's analyze each part:

  1. The Numerator's First Term: In the general formula, we have −b-b. In Haley's work, we see −(−10)-(-10). This tells us that −b=−(−10)-b = -(-10), which simplifies to b=−10b = -10.
  2. The Discriminant: The discriminant in the quadratic formula is b2−4acb^2 - 4ac. In Haley's expression, we have (−10)2−4(7)(−2)(-10)^2 - 4(7)(-2). By comparing these, we can identify a and c. We see that a corresponds to 7 and c corresponds to -2.
  3. The Denominator: The denominator in the quadratic formula is 2a2a. In Haley's work, the denominator is 2(7)2(7). This confirms our earlier finding that a=7a = 7.

Therefore, by meticulously comparing Haley's work with the quadratic formula, we have identified the coefficients: a = 7, b = -10, and c = -2.

Constructing the Quadratic Function in Standard Form

Now that we have determined the values of a, b, and c, we can easily write the quadratic function in standard form:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

Substituting the values we found:

f(x)=7x2+(−10)x+(−2)f(x) = 7x^2 + (-10)x + (-2)

Simplifying, we get:

f(x)=7x2−10x−2f(x) = 7x^2 - 10x - 2

This is the quadratic function in standard form that Haley used to arrive at her solution in the quadratic formula.

Verification and Deeper Understanding

To ensure our solution is correct, we can substitute the values of a, b, and c back into the quadratic formula and see if we arrive at Haley's expression. This step reinforces our understanding of the relationship between the quadratic equation and its solutions.

Substituting a = 7, b = -10, and c = -2 into the quadratic formula:

x=−(−10)±(−10)2−4(7)(−2)2(7)x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(7)(-2)}}{2(7)}

This is exactly the expression Haley obtained, confirming that our solution is correct.

The Significance of the Discriminant

As we mentioned earlier, the discriminant (b2−4acb^2 - 4ac) plays a crucial role in determining the nature of the roots of a quadratic equation. In this case, the discriminant is:

(−10)2−4(7)(−2)=100+56=156(-10)^2 - 4(7)(-2) = 100 + 56 = 156

Since the discriminant is positive, we know that the quadratic equation has two distinct real roots. This means the parabola representing the quadratic function intersects the x-axis at two different points. Understanding the discriminant adds another layer of insight into the behavior of quadratic functions.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used in physics to model projectile motion, in engineering to design bridges and structures, and in finance to calculate growth rates and compound interest. Understanding quadratic equations and their solutions is essential in many fields of study and practical applications.

For instance, imagine a ball thrown into the air. Its trajectory can be modeled by a quadratic equation, where the height of the ball at any given time is a function of time. The quadratic formula can then be used to determine when the ball will hit the ground (i.e., when the height is zero). This is just one example of how quadratic equations are used to solve real-world problems.

Common Mistakes and How to Avoid Them

When working with the quadratic formula, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  1. Sign Errors: One of the most common errors is making mistakes with signs, especially when dealing with negative values of b and c. Always double-check your signs and be careful when substituting values into the formula.
  2. Order of Operations: Remember to follow the correct order of operations (PEMDAS/BODMAS) when simplifying the expression under the square root (the discriminant). First, calculate the square, then the multiplication, and finally the subtraction.
  3. Incorrect Substitution: Ensure that you substitute the correct values for a, b, and c. A simple mistake in substitution can lead to a completely wrong answer.
  4. Simplifying the Radical: After calculating the discriminant, you may need to simplify the square root. Make sure you simplify it completely by factoring out any perfect squares.

By paying close attention to these potential pitfalls, you can improve your accuracy and confidence when using the quadratic formula.

Conclusion: Mastering Quadratic Equations

In this article, we successfully reversed Haley's steps in using the quadratic formula to identify the original quadratic function in standard form. We achieved this by carefully comparing her work with the general formula and identifying the coefficients a, b, and c. We then constructed the quadratic function f(x)=7x2−10x−2f(x) = 7x^2 - 10x - 2.

This exercise highlights the importance of understanding the quadratic formula and its connection to the coefficients of a quadratic equation. By mastering these concepts, you will be well-equipped to solve a wide range of problems involving quadratic equations. Remember to practice regularly and pay attention to detail to avoid common mistakes. The journey to mastering quadratic equations is a rewarding one, opening doors to a deeper understanding of mathematics and its applications in the real world.

Understanding the nuances of the quadratic formula and how it relates to the coefficients of a quadratic equation in standard form is a fundamental skill in algebra. The standard form, represented as f(x)=ax2+bx+cf(x) = ax^2 + bx + c, provides a clear structure for identifying the key parameters that dictate the behavior of the quadratic function. In this case, the ability to deconstruct Haley's application of the quadratic formula demonstrates a strong grasp of the underlying principles. By carefully comparing each term in Haley's work with the general form of the quadratic formula, we were able to accurately extract the values of a, b, and c. This process not only provides a solution to the specific problem but also reinforces the ability to analyze and interpret mathematical expressions. The correct identification of a = 7, b = -10, and c = -2 is crucial for reconstructing the original quadratic function. This exercise emphasizes the importance of attention to detail and a systematic approach to problem-solving in mathematics. The quadratic formula is a versatile tool that can be applied to a wide range of problems, making it an essential concept for students to master. The ability to not only apply the formula but also to reverse-engineer its application, as we did in this article, showcases a deeper understanding of the underlying mathematical principles. This understanding allows for greater flexibility and problem-solving prowess when dealing with quadratic equations and their applications. The process of verification, by substituting the values back into the quadratic formula, is a critical step in ensuring the accuracy of the solution. This step highlights the importance of checking one's work and reinforces the connection between the coefficients and the solutions of the equation. By substituting a = 7, b = -10, and c = -2 back into the quadratic formula, we confirmed that our reconstructed quadratic function aligns perfectly with Haley's initial expression. This verification step adds a layer of confidence to the solution and demonstrates a thorough understanding of the mathematical relationships involved. The positive value of the discriminant in this problem indicates that the quadratic equation has two distinct real roots. This insight allows for a deeper understanding of the behavior of the quadratic function, specifically its intersections with the x-axis. The discriminant, calculated as 156, provides valuable information about the nature of the roots without the need to explicitly solve for them. Understanding the significance of the discriminant is a key aspect of mastering quadratic equations and their properties. This understanding enhances the ability to analyze and interpret quadratic functions in various contexts. The real-world examples of quadratic equations, such as projectile motion and financial modeling, underscore the practical relevance of this mathematical concept. By understanding how quadratic equations are used in diverse fields, students can appreciate the broader applicability of their mathematical knowledge. The trajectory of a ball thrown into the air, for instance, can be accurately modeled using a quadratic equation, allowing for predictions about its height and landing point. This connection to real-world phenomena makes the study of quadratic equations more engaging and meaningful. Addressing common mistakes, such as sign errors and incorrect substitutions, is an essential part of effective learning. By identifying and avoiding these pitfalls, students can improve their accuracy and problem-solving efficiency. Sign errors, in particular, are a frequent source of mistakes when working with the quadratic formula. Careful attention to detail and a systematic approach can help minimize these errors. The conclusion of this article reinforces the importance of practice and attention to detail in mastering quadratic equations. The journey to understanding quadratic functions and their applications is a continuous process that requires consistent effort and a willingness to learn from mistakes. By embracing this approach, students can develop a strong foundation in algebra and unlock a wide range of problem-solving capabilities. The ability to dissect complex mathematical expressions and reverse-engineer solutions, as demonstrated in this article, is a valuable skill that extends beyond the realm of quadratic equations.

Answer

f(x)=7x2−10x−2f(x) = 7x^2 - 10x - 2