Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving the equation $y = ax^2 + c$ for $x$. This is a common task in algebra, and understanding how to manipulate this equation is super helpful. We will go through the process, step-by-step, to make sure everyone gets it. I will then break down the multiple-choice options to help you select the correct answer. Let's get started!

Isolating the $x^2$ Term

The first thing we need to do is isolate the term containing $x^2$. To do this, we'll start by subtracting $c$ from both sides of the equation. This keeps the equation balanced. You get:

$y - c = ax^2$

See? Simple! We've successfully moved the constant term to the other side, and now $ax^2$ is on its own. This is a critical step because we are trying to get $x$ by itself. Understanding the order of operations (PEMDAS/BODMAS) is super important here. We have to work in reverse order to undo the operations that have been applied to x.

Now, we move onto our second step. It is imperative to maintain the equality throughout the entire process. Always perform the same operations on both sides. This is a fundamental principle of algebra, ensuring that the solution is correct. By doing this, you're essentially "undoing" the operations that have been applied to $x$. We are not changing the value of x, but simply rewriting the equation in a way that makes it easier to solve for x. Remember that we want to isolate $x$ on one side of the equation.

Think of it like unwrapping a present. You have to undo each layer to get to the final item. The same logic applies to equations. Each step we perform removes a layer, bringing us closer to the solution. This step-by-step method helps to break down complex problems into a series of manageable tasks. This strategy makes complex problems easier to tackle. So far, we have only done one step, but it is important to note that this is not a race. We are making sure that each step is correct.

Solving for $x$

Our next goal is to get $x$ by itself. After subtracting $c$ from both sides, our equation looks like $y - c = ax^2$. We can divide both sides by $a$. This leaves us with $x^2$ all by itself on one side:

$\frac{y - c}{a} = x^2$

Awesome! Now, we just need to get rid of the square. How do we do that? By taking the square root of both sides!

Remember, when you take the square root, you have to consider both the positive and negative solutions. That means we'll get two possible values for $x$.

So, the equation becomes: $x = \pm \sqrt{\frac{y - c}{a}}$

The plus-or-minus symbol ($\pm$) indicates that there are two possible solutions: one positive and one negative. This is because both a positive and a negative number, when squared, result in a positive number. Recognizing this is critical. When solving equations, always keep in mind the different possibilities that can exist. Do not assume an answer. Think critically, and evaluate the different possible answers to avoid the pitfalls. Think through the different algebraic manipulations to make sure the answer is correct. The square root operation is the inverse of squaring, so taking the square root of $x^2$ gives us $x$. Therefore, when taking the square root, it is essential to consider both positive and negative roots to account for all possible solutions.

We are now at the last step, so let us appreciate the journey we have taken. We have been working diligently throughout all the steps, so let's finish strong. Always double-check to make sure there are no mistakes. Keep in mind that mathematical problems can be tricky. There may be instances where it can be easy to get lost. Break down the problems to make them easier to solve. Keep going, and never give up!

Analyzing the Multiple-Choice Options

Let's break down the multiple-choice options to see which one matches our solution.

• Option A: $x= \pm \sqrt{ay - c}$ This option is incorrect because it incorrectly multiplies a by y before subtracting c. The correct operation requires subtracting c first and then dividing by a.

• Option B: $x = \pm \sqrt{\frac{y - c}{a}}$ This is the correct answer! It matches the solution we derived step-by-step. The equation is correctly solved for x, accounting for both positive and negative square roots.

• Option C: $x = \sqrt{\frac{y}{a} - c}$ This option is incorrect because it doesn't consider the negative root. Additionally, it correctly divides y by a but then incorrectly subtracts c, which should be within the square root after dividing by a.

• Option D: $x = \sqrt{\frac{y + c}{a}}$ This option is incorrect because it adds c instead of subtracting it, and it doesn't account for the negative root. The signs are wrong, and it is missing the plus or minus sign. Therefore, this option is also incorrect. The numerator is off, and it is missing the plus-or-minus sign, which is very important.

Conclusion

Alright, guys, we've solved the equation and found the correct answer! The key steps were isolating the $x^2$ term, dividing by a, and then taking the square root of both sides, remembering the plus-or-minus. I hope this helped you all. Remember to practice similar problems to solidify your understanding. Keep up the great work. Math can be fun, I swear!

In summary: The correct answer is Option B: $x = \pm \sqrt{\frac{y - c}{a}}$. This is the final solution to the equation $y = ax^2 + c$ for $x$. We have solved for x step by step. Now you can use it for any problems you may encounter in the future.

I hope this comprehensive guide has been helpful. Always remember to take your time, understand each step, and practice! Keep up the good work!