Solving 3x² - 2 = 0: Step-by-Step Solutions

by Sebastian Müller 44 views

Hey guys! Ever stumbled upon a math problem that just seems to throw you for a loop? Well, you're not alone! Math equations can sometimes feel like a puzzle, but with the right approach, we can crack them open and find the solutions we're looking for. Today, we're going to dive into solving a specific equation: 3x² - 2 = 0. This is a classic quadratic equation, and we'll break down the steps to find its solutions in a way that's super easy to understand.

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what a quadratic equation is. In essence, a quadratic equation is a polynomial equation of the second degree. That probably sounds like a mouthful, but it simply means it's an equation where the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. Our equation, 3x² - 2 = 0, fits this form, with a = 3, b = 0 (since there's no 'x' term), and c = -2.

Now, why are we even bothering with quadratic equations? Well, they pop up all over the place in real-world applications. Think about projectile motion in physics, calculating areas, or even modeling growth and decay in biology. Knowing how to solve them is a valuable skill to have in your mathematical toolkit. So, let's get to it!

Methods for Solving Quadratic Equations

There are several methods we can use to solve quadratic equations, each with its own strengths and weaknesses. The most common methods include:

  1. Factoring: This method involves breaking down the quadratic expression into a product of two binomials. It's often the quickest method when it works, but it's not always applicable.

  2. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side. It's a more general method than factoring and can be used to solve any quadratic equation.

  3. Quadratic Formula: This is the most versatile method, as it can be used to solve any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is given by:

    x = (-b ± √(b² - 4ac)) / 2a

For our equation, 3x² - 2 = 0, the quadratic formula is the most straightforward method to use, but we'll also explore a simpler approach due to the absence of the 'x' term.

Solving 3x² - 2 = 0: A Step-by-Step Guide

Okay, let's tackle our equation head-on! 3x² - 2 = 0. Notice that this equation is a bit simpler than the general form because it's missing the 'bx' term. This means we can use a more direct approach to find the solutions.

Step 1: Isolate the x² Term

Our first goal is to get the x² term by itself on one side of the equation. To do this, we'll add 2 to both sides:

3x² - 2 + 2 = 0 + 2

This simplifies to:

3x² = 2

Step 2: Divide by the Coefficient of x²

Now, we want to get x² completely alone, so we'll divide both sides of the equation by 3:

3x² / 3 = 2 / 3

This gives us:

x² = 2 / 3

Step 3: Take the Square Root of Both Sides

To solve for x, we need to get rid of the square. The inverse operation of squaring is taking the square root. So, we'll take the square root of both sides:

√(x²) = ±√(2 / 3)

Remember that when we take the square root of a number, we need to consider both the positive and negative roots. This is because both a positive and a negative number, when squared, will result in a positive number.

Step 4: Simplify the Square Root

Now, let's simplify the square root. We have:

x = ±√(2 / 3)

To get rid of the square root in the denominator, we can multiply the numerator and denominator inside the square root by 3:

x = ±√(2 * 3 / 3 * 3)

x = ±√(6 / 9)

Now we can take the square root of the denominator:

x = ±√6 / √9

x = ±√6 / 3

So, our two solutions are:

x = √6 / 3 and x = -√6 / 3

Identifying the Correct Alternative

Now that we've found the solutions, let's match them to the alternatives provided. Looking at the options, we need to find the one that includes both √6 / 3 and -√6 / 3. The correct alternative is the one that matches these values. It's crucial to remember the ± sign, which indicates the presence of both positive and negative roots.

Why are there two solutions?

You might be wondering, why do we get two solutions for a quadratic equation? Well, graphically, a quadratic equation represents a parabola. The solutions to the equation are the points where the parabola intersects the x-axis. A parabola can intersect the x-axis at two points, one point, or no points, depending on the equation. In our case, the parabola intersects the x-axis at two points, hence the two solutions.

Common Mistakes to Avoid

When solving quadratic equations, there are a few common mistakes that students often make. Let's go over them to help you steer clear of these pitfalls:

  • Forgetting the ± Sign: This is a big one! When taking the square root of both sides of an equation, always remember to include both the positive and negative roots. Otherwise, you'll miss one of the solutions.
  • Incorrectly Simplifying Square Roots: Make sure you simplify square roots correctly. Remember that √(a / b) is equal to √a / √b, and you can often simplify further by finding perfect square factors.
  • Misapplying the Quadratic Formula: The quadratic formula is a powerful tool, but it's easy to make a mistake if you don't plug in the values correctly. Double-check your substitutions and be careful with the signs.
  • Skipping Steps: It's tempting to rush through the steps, but skipping steps can lead to errors. Take your time and write out each step clearly.

Tips for Mastering Quadratic Equations

Want to become a quadratic equation whiz? Here are a few tips to help you master these types of problems:

  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with solving quadratic equations. Work through a variety of problems, and don't be afraid to make mistakes – that's how you learn!
  • Understand the Concepts: Don't just memorize formulas; try to understand the underlying concepts. This will help you apply the methods correctly and solve problems more efficiently.
  • Check Your Answers: After you've found a solution, plug it back into the original equation to make sure it works. This is a great way to catch errors.
  • Use Online Resources: There are tons of great resources online that can help you learn about quadratic equations, including videos, tutorials, and practice problems. Websites like Khan Academy and Wolfram Alpha are excellent starting points.
  • Don't Be Afraid to Ask for Help: If you're struggling with quadratic equations, don't hesitate to ask your teacher, a tutor, or a classmate for help. We all need a little help sometimes!

Conclusion: You've Got This!

So, there you have it! We've walked through the process of solving the equation 3x² - 2 = 0 step-by-step, identified the correct alternative, and discussed common mistakes and tips for mastering quadratic equations. Remember, math can be challenging, but with a little effort and the right approach, you can conquer any equation that comes your way. Keep practicing, stay curious, and you'll be solving quadratic equations like a pro in no time! You got this, guys!