Solving Quadratic Equations Step By Step A Detailed Analysis

Hey everyone! Today, we're diving into a quadratic equation problem that Jessica tackled. She was tasked with finding the equation of a quadratic function that passes through the point (8, -11) and has its vertex at (6, -3). Let's break down her work step-by-step, see what she did right, and maybe spot some areas where we can learn and improve. Understanding quadratic functions is super important in math, as they pop up everywhere from physics to engineering. So, let's get started and make sure we're all on the same page with this!

Jessica's Initial Steps: Setting Up the Vertex Form

Jessica kicked things off by using the vertex form of a quadratic equation, which is a fantastic approach for this problem. The vertex form is given by:

f(x) = a(x - h)^2 + k

Where:

• (h, k) is the vertex of the parabola
• a determines the direction and steepness of the parabola

Jessica correctly identified that the vertex (h, k) is (6, -3). Plugging these values into the vertex form, we get:

f(x) = a(x - 6)^2 - 3

Now, to find the value of 'a', which tells us how much the parabola stretches or compresses, we need to use the other piece of information we have: the point (8, -11) that the function passes through. This means when x = 8, f(x) = -11. Jessica did a great job setting up the equation in her first step by substituting these values:

-11 = a(8 - 6)^2 - 3

This is the crucial first step, and Jessica nailed it! She correctly substituted the given point and vertex into the vertex form of the quadratic equation. This equation now allows us to solve for the unknown 'a', which will complete our quadratic function definition. This initial setup demonstrates a solid understanding of how the vertex form works and how to apply given information. By using the vertex form, Jessica has streamlined the process of finding the equation, making it easier to solve for the remaining unknown. This is a testament to her grasp of key quadratic equation concepts and her ability to apply them effectively.

Solving for 'a': A Closer Look at Jessica's Calculations

The next few steps in Jessica's work involve solving for the value of 'a'. This is where the algebraic manipulation comes into play, and it's essential to follow each step carefully to ensure accuracy. Let's trace her calculations:

1. -11 = a(8 - 6)^2 - 3 (This is the equation we established in the previous step)
2. -11 = a(2)^2 - 3 (Jessica correctly simplified the expression inside the parenthesis)
3. -11 = 4a - 3 (Squaring 2 gives us 4, so this step is also correct)
4. -8 = 4a (Adding 3 to both sides to isolate the term with 'a' is the right move)
5. a = -2 (Finally, dividing both sides by 4 to solve for 'a')

Looking at these steps, Jessica has executed the algebraic operations flawlessly. Each step logically follows from the previous one, demonstrating a solid understanding of equation solving. The simplification, isolation of the variable, and final division are all performed accurately. The quadratic equation skill shown here is impressive, as each step is crucial in reaching the correct solution. Solving for 'a' is a critical part of defining the quadratic function, as it determines the shape and direction of the parabola. Jessica's meticulous calculations here are a great example of the attention to detail needed in algebra. The final value of a = -2 is a key piece of the puzzle, telling us that the parabola opens downwards and is stretched vertically.

Constructing the Quadratic Equation: Putting It All Together

Now that we've found the value of 'a', we can construct the complete quadratic equation. Remember, we started with the vertex form:

f(x) = a(x - h)^2 + k

We know:

• a = -2
• h = 6 (from the vertex)
• k = -3 (from the vertex)

Substituting these values, we get:

f(x) = -2(x - 6)^2 - 3

This is the quadratic equation in vertex form that represents the function passing through the point (8, -11) and having a vertex at (6, -3). Jessica successfully found this equation by correctly applying the vertex form and solving for 'a'. But, just for completeness, let's expand this equation to get it into the standard form, which is:

f(x) = Ax^2 + Bx + C

Expanding our vertex form equation:

f(x) = -2(x^2 - 12x + 36) - 3 f(x) = -2x^2 + 24x - 72 - 3 f(x) = -2x^2 + 24x - 75

So, the quadratic equation in standard form is f(x) = -2x^2 + 24x - 75. This confirms that Jessica's solution is indeed correct, and she has a firm grasp on both the vertex form and the standard form of quadratic equations. The ability to convert between these forms is a valuable skill in mathematics, allowing for different perspectives on the same function. This comprehensive understanding of quadratic functions is a testament to Jessica's mathematical abilities.

Common Pitfalls and How to Avoid Them

Working with quadratic equations can sometimes be tricky, and there are a few common pitfalls that students often encounter. Recognizing these potential errors can help us avoid them in the future. One common mistake is mixing up the signs when substituting the vertex coordinates (h, k) into the vertex form. Remember, the formula is f(x) = a(x - h)^2 + k, so the x-coordinate of the vertex (h) is subtracted. Another frequent error is in the algebraic manipulation while solving for 'a'. It's crucial to follow the order of operations (PEMDAS/BODMAS) and double-check each step to avoid mistakes in simplification or isolating the variable. For example, distributing the 'a' value after squaring the (x - h) term requires careful attention to signs and coefficients. A small error in arithmetic can lead to an incorrect value of 'a', and consequently, the wrong quadratic equation.

Another area where mistakes can happen is in the expansion of the vertex form to the standard form. Squaring the binomial (x - h) and then distributing the 'a' value requires careful multiplication and combining like terms. A missed sign or incorrect multiplication can throw off the entire equation. To avoid these pitfalls, it's always a good idea to write out each step clearly and double-check your work. Practice is also key – the more you work with quadratic equations, the more comfortable you'll become with the process and the less likely you are to make mistakes. Additionally, using graphing calculators or online tools to verify your solutions can be a helpful way to catch errors and build confidence in your answers. By being aware of these common pitfalls and actively working to avoid them, you can master quadratic equations and tackle even the most challenging problems with ease. Understanding quadratic functions is more than just memorizing formulas; it's about developing a strong conceptual understanding and the ability to apply that understanding to solve problems accurately and efficiently.

Real-World Applications of Quadratic Equations

Quadratic equations aren't just abstract math concepts; they have tons of real-world applications! You might be surprised at how often they pop up in everyday life and various fields. One classic example is in physics, where quadratic equations are used to describe the trajectory of projectiles, like a ball thrown in the air or a rocket launched into space. The parabolic path that these objects follow can be modeled using a quadratic function, allowing us to predict their height and distance traveled. Understanding quadratic equations is essential for physicists and engineers who work with motion and trajectory.

In engineering, quadratic equations are used in the design of bridges, arches, and other structures. The parabolic shape is incredibly strong and efficient for distributing weight, making it a popular choice for these types of constructions. Quadratic equations also play a role in optimizing designs for acoustics and aerodynamics. In economics and finance, quadratic functions can be used to model cost, revenue, and profit. For example, a company might use a quadratic equation to determine the optimal price point for a product to maximize profit. Understanding these applications helps to see the practical relevance of quadratic functions beyond the classroom. Moreover, quadratic equations are used in computer graphics and video game development to create realistic curves and shapes. The parabolic curves generated by quadratic functions are used to model lighting effects, object shapes, and movement paths. This highlights the versatility of quadratic equations and their importance in various technological fields. The ability to apply quadratic equations to solve real-world problems is a valuable skill that extends far beyond mathematics, making it a fundamental concept in many disciplines. So, the next time you see a curved structure or a projectile in motion, remember that quadratic equations are likely at play, shaping the world around us.

Conclusion: Mastering Quadratic Equations

In conclusion, Jessica's journey to find the quadratic equation is a great example of how to approach these types of problems. She correctly used the vertex form, substituted the given point, and solved for 'a' accurately. By understanding the vertex form and how to manipulate it, we can confidently tackle similar challenges. Remember, quadratic equations are fundamental in mathematics and have numerous real-world applications. From physics to engineering to economics, they help us model and understand the world around us. So, mastering quadratic equations is not just about getting good grades; it's about developing a powerful tool for problem-solving in various fields. The ability to identify, analyze, and solve quadratic equations is a valuable skill that will serve you well in your academic and professional pursuits.

By breaking down the problem step-by-step, we can see the logic and reasoning behind each action. This approach not only helps us understand the solution but also builds confidence in our own problem-solving abilities. Whether you're a student learning quadratic equations for the first time or someone looking to refresh your knowledge, remember to focus on the fundamentals, practice regularly, and don't be afraid to ask questions. With a solid understanding of the concepts and a bit of practice, you can master quadratic equations and unlock their potential for solving real-world problems. Keep exploring, keep learning, and keep pushing your mathematical boundaries! Remember, the journey of learning mathematics is continuous, and every step you take brings you closer to mastery.