Solving For X: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for x in the equation 3x² + 6 = 3. Don't worry if quadratic equations sound intimidating; we'll break it down into easy-to-follow steps. This guide is designed to help anyone understand how to isolate x and find the solution. Whether you're a student tackling homework or just brushing up on your math skills, this tutorial will provide a clear, concise path to the answer. Let's get started, shall we?

Understanding the Basics: Quadratic Equations

Alright, before we jump into the nitty-gritty, let's chat about what we're dealing with. The equation 3x² + 6 = 3 is a type of equation called a quadratic equation. Quadratic simply means that the equation has a term where the variable (x in our case) is raised to the power of 2 (x²). These equations typically have two solutions, or roots, which are the values of x that make the equation true. In our case, the goal is to find those x values. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. Notice that our equation 3x² + 6 = 3 is not yet in this standard form, but we'll get there. Knowing this helps us understand the structure of the problem and the general approach we'll take. We're essentially working backward to isolate x and reveal its value(s). The journey involves using inverse operations to simplify the equation and get x by itself. This process ensures we maintain the equality and arrive at the correct solutions. Furthermore, understanding the properties of quadratic equations will assist you to solve more complex equations. Ready to become math ninjas?

To solve a quadratic equation, we'll generally go through a few key steps: First, we need to arrange the equation into standard form. That means getting all the terms on one side of the equation and zero on the other side. This is crucial as it helps us identify the coefficients a, b, and c in the equation, which we might need for later steps like using the quadratic formula or factoring. Second, we can try to factor the quadratic expression if possible. Factoring is like breaking down the equation into simpler parts. If the quadratic expression is factorable, this can be a quick way to find the solutions. Finally, if factoring isn't straightforward, we can use the quadratic formula. This formula is a magical tool that works for any quadratic equation, regardless of how complicated it looks. The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It's the ultimate problem-solver when it comes to quadratic equations. Throughout this solution, we will demonstrate each of these steps and provide insights to simplify solving for x.

Step-by-Step Solution: Isolating x

Alright, let's get down to business! Here's how we'll solve 3x² + 6 = 3 step-by-step:

Step 1: Simplify the Equation

Our first order of business is to get the equation into a more manageable form. The goal is to isolate the x² term. We'll start by subtracting 6 from both sides of the equation. This keeps everything balanced and allows us to move terms around.

3x² + 6 - 6 = 3 - 6

This simplifies to:

3x² = -3

See? Already looking better! We've removed the constant term from the left side, bringing us closer to isolating x. Remember, every operation we perform on one side of the equation must be done on the other to maintain equality. This is the golden rule of algebra. This step is pretty critical because it sets the stage for the next steps. It allows us to focus on the term with x in it, making it easier to solve for x. This simplification will make the subsequent steps less complex and less prone to errors.

Step 2: Isolate x²

Now, we need to get x² by itself. To do this, we'll divide both sides of the equation by 3. This cancels out the coefficient in front of x², leaving us with just x².

3x² / 3 = -3 / 3

This simplifies to:

x² = -1

Great! We've successfully isolated x². Now, we are ready to move on the final step to find the value of x. The equation is now much simpler. By eliminating the coefficient of the x² term, we've brought us closer to the solution. The simplified form of the equation is much easier to work with, making the next step straightforward.

Step 3: Solve for x

Almost there! To find the value of x, we need to take the square root of both sides of the equation. Remember, when you take the square root, you have to consider both the positive and negative square roots. This means there will usually be two possible solutions for x.

√(x²) = ±√(-1)

This gives us:

x = ±√(-1)

Now, here's where things get interesting. The square root of -1 is not a real number. It's an imaginary number, often denoted as i. So, √(-1) = i. Our solutions are therefore:

x = i and x = -i

And that's it! We've solved for x. The solutions to the equation 3x² + 6 = 3 are x = i and x = -i. These are complex solutions, which means they involve the imaginary unit i. Taking the square root of both sides is a crucial step in isolating x because it allows us to eliminate the exponent. Recognizing that we're dealing with the square root of a negative number is important, because this tells us the solutions are complex. Also remember, quadratic equations can have two solutions, one solution, or no real solutions, depending on the equation and the values involved.

Conclusion: Wrapping Up the Solution

So, there you have it, guys! We've successfully solved the quadratic equation 3x² + 6 = 3. The solutions, x = i and x = -i, are complex numbers. Solving this kind of equation involves simplifying the equation, isolating the x² term, and then taking the square root of both sides. Sometimes, as we saw here, the solutions might not be real numbers. This step-by-step guide is designed to provide you with the necessary tools to tackle any quadratic equation. We’ve covered everything from basic principles to the final solution, ensuring a solid understanding of the concepts involved. Practicing similar problems will further solidify your knowledge. Keep at it, and you'll be a pro in no time.

Remember, mastering algebra takes practice. Don't be afraid to work through more examples and ask questions if you get stuck. The more you practice, the more comfortable you'll become with solving quadratic equations. Keep up the great work!