Solving Quadratic Equations: Is This Solution Right?

by Dimemap Team 53 views

Hey guys! Let's dive into a common math problem: solving quadratic equations. Specifically, we're going to examine a proposed solution and figure out if it's correct. This is super important because understanding how to solve these equations is a fundamental skill in algebra, and it pops up in all sorts of other math and science fields. So, buckle up, and let's break down the steps and see if the answer holds water!

The Problem and the Attempted Solution

Alright, here's the quadratic equation we're working with:

9x + 2 = 8x^2 + 6x

The person solving this equation went through the following steps. Let's take a look:

  1. Rearranging the equation: -8x^2 + 3x + 2 = 0
  2. Using the quadratic formula: x = (-3 ± √(3^2 - (4)(-8)(2))) / -16
  3. Simplifying the discriminant: x = (-3 ± √(9 - 64)) / -16
  4. Calculating the final answer: x = (3 ± √55i) / 16

Looks pretty involved, right? Now, let's dissect this step-by-step to check for any potential errors and see if the solution is valid.

Step-by-step Analysis

Let's meticulously go through each step of this solution to see if it's on the right track. We'll be on the lookout for mistakes in the rearrangement, in the application of the quadratic formula, and in the arithmetic.

First, we need to bring all terms to one side to get a standard quadratic equation form ax^2 + bx + c = 0. The first step, -8x^2 + 3x + 2 = 0, seems correct. They've rearranged the initial equation by subtracting 9x and 2 from both sides, and then subtracting 6x from both sides and reordered, which results in subtracting 6x to get 3x. This is a crucial step because it sets the stage for using the quadratic formula.

Next, the solver applied the quadratic formula. The quadratic formula is a lifesaver, and it's used to find the solutions (also called roots) of a quadratic equation. The formula is x = (-b ± √(b^2 - 4ac)) / 2a. Here, a, b, and c are the coefficients from our standard quadratic equation ax^2 + bx + c = 0. For our equation, -8x^2 + 3x + 2 = 0, we have a = -8, b = 3, and c = 2. Let's compare this with the provided solution. It is x = (-3 ± √(3^2 - (4)(-8)(2))) / -16. The formula is correctly applied, but the denominator is wrong. It should be 2a = 2*(-8) = -16. So far, so good, but we need to pay close attention to the details to ensure accuracy.

Let's get into the simplification. In the third step, the solver simplified the discriminant which is b^2 - 4ac. The discriminant inside the square root is correctly calculated as 3^2 - (4)(-8)(2) = 9 - (-64) = 9 + 64 = 73. But in the provided solution the discriminant is simplified to 9 - 64 = -55, which is wrong.

Finally, the fourth step involves the square root of a negative number, which results in an imaginary number. However, because of the error in the discriminant, the result is also wrong. The correct answer should be x = (-3 ± √73) / -16. The solution provided gave x = (3 ± √55i) / 16, which is not correct. The errors in the discriminant result in a final incorrect solution, and so, the solution is invalid.

Identifying the Mistakes: Where Things Went Wrong

Let's pinpoint exactly where the solution went off the rails. The solver made two key errors.

  1. Error in the Discriminant Calculation: The most critical mistake happened when calculating the discriminant (the part inside the square root in the quadratic formula). They incorrectly calculated 3^2 - (4)(-8)(2) as 9 - 64, instead of correctly calculating it as 9 - (-64) = 9 + 64 = 73. This arithmetic error is fundamental because it changes the nature of the roots.
  2. Incorrect Arithmetic in the Final Solution: The final solution was also incorrect because of the error in the discriminant calculation. The correct final solution must be derived using the correct discriminant value. The wrong value caused a cascading error that led to an incorrect answer.

These errors led to an incorrect answer. The errors in the calculation of the discriminant lead to the wrong answer. Understanding where these errors occurred is key to improving our problem-solving skills and avoiding similar pitfalls in the future.

The Correct Solution and Explanation

Let's work through the correct solution to ensure we understand how to solve the quadratic equation. Starting from -8x^2 + 3x + 2 = 0, we will use the quadratic formula.

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. Where, in our equation, a = -8, b = 3, and c = 2.

Substituting these values, we get:

x = (-3 ± √(3^2 - 4 * (-8) * 2)) / (2 * -8)

Simplifying further:

x = (-3 ± √(9 + 64)) / -16

x = (-3 ± √73) / -16

Therefore, the correct solutions are:

x = (-3 + √73) / -16 and x = (-3 - √73) / -16

So, the correct solution involves the correct use of the quadratic formula, and careful arithmetic. It also highlights the importance of the correct discriminant.

Key Takeaways and Conclusion

This exercise highlights several crucial points about solving quadratic equations. First, mastering the quadratic formula is essential. Second, meticulous arithmetic is non-negotiable! Every step, from simplifying the discriminant to the final calculation, needs to be done carefully. Third, a good understanding of the standard form of quadratic equations is a must. Knowing how to manipulate the equation into the standard form is the first step to the solution.

So, to answer the question: Is the solution shown below correct? No, it's not. The incorrect calculation of the discriminant and the final answer made the solution invalid. However, by thoroughly analyzing the steps and identifying the errors, we've strengthened our understanding of how to solve these equations correctly and how to avoid common mistakes.

By carefully working through the problem step by step, we can catch those errors and make sure that we get the right answer! Keep practicing, guys, and you'll become quadratic equation masters in no time!

Let's add some extra tips to your math toolbox to help you conquer quadratic equations with confidence. These tips will not only help you get the right answers but also boost your understanding of the underlying principles. Ready? Let's go!

1. Practice Makes Perfect: Consistent Practice is Key

First, like any skill, solving quadratic equations improves with practice. The more you work through problems, the more familiar you'll become with the formulas and the patterns. Start with simple problems and gradually move to more complex ones. Make sure to solve a variety of problems that represent all the forms of the equation.

Try different methods like the factoring method, completing the square, or using the quadratic formula. Each method offers a different way to approach the problem, and practicing all of them enhances your overall mathematical flexibility. Practice makes it easier to catch errors and makes each step of the process faster.

2. Master the Quadratic Formula and Its Components

The quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a is your best friend when it comes to quadratic equations. Understand each component of the formula: a, b, and c. Remember that the values of a, b, and c are derived from the standard form of the equation: ax^2 + bx + c = 0. Misidentifying these coefficients is a common mistake that can lead to wrong answers, so make sure to get this part right!

Also, pay close attention to the discriminant, b^2 - 4ac. The discriminant gives us valuable information about the nature of the roots (solutions) of the equation. If the discriminant is positive, you have two real solutions. If it's zero, you have one real solution. If it's negative, you have two complex solutions. Knowing the characteristics of the roots before you calculate them can help you catch errors early.

3. Double-Check Your Calculations: Accuracy Matters

Arithmetic errors are your enemy. Always double-check your calculations, especially when simplifying the discriminant and applying the quadratic formula. A small mistake in arithmetic can lead to a completely incorrect solution. Use a calculator to double-check, but make sure you understand each step before relying on technology.

Write down each step clearly and neatly. This will help you track your work and identify mistakes more easily. Avoid doing too much mental math. Write out each step, and you'll find it easier to find and fix errors.

4. Understand Different Methods: Expand Your Toolkit

While the quadratic formula is a reliable method, understanding other methods like factoring and completing the square can be beneficial. Factoring is useful when the equation can be easily factored, which can make the solution faster. Completing the square is another approach, especially when the equation cannot be easily factored, or you are looking to rewrite the equation in a different form. Each method has its advantages, and knowing multiple methods enhances your problem-solving skills.

Practicing different methods also reinforces your understanding of the underlying algebraic concepts. It shows you the relationships between different parts of the equation, and builds a deeper understanding than just relying on one formula.

5. Simplify and Organize: Keep It Clean

Simplify the equation as much as possible before starting to solve it. This includes combining like terms and reducing coefficients. A simpler equation is less prone to errors. Clear organization helps you avoid mistakes. Write out the steps clearly and label them properly. This not only helps you solve the problem more efficiently but also makes it easier to review your work and identify any errors.

6. Check Your Answers: Verification Is Crucial

Always check your answers! Substitute your solutions back into the original equation to make sure they satisfy the equation. This is the ultimate test of whether you've found the correct solutions. If the equation holds true after substitution, your solution is correct. If not, go back and review your steps to find the mistake.

7. Seek Help When Needed: Don't Be Afraid to Ask

If you're struggling with a problem, don't hesitate to ask for help. Ask a teacher, a tutor, or a classmate. Explaining your problem to someone else can often help you clarify your own understanding. It can also provide a fresh perspective and help you catch mistakes you might have missed.

There are tons of online resources like video tutorials, online forums, and practice quizzes. Using these resources can offer alternative explanations and plenty of practice problems.

By following these tips, you'll not only improve your ability to solve quadratic equations but also boost your overall mathematical confidence. Good luck, and keep practicing!