Solving $3x^2-7=0$ By The Square Root Property A Step By Step Guide

Let's dive into solving the equation $3x^2 - 7 = 0$ using the square root property. This method is super handy when you've got a squared term isolated on one side of the equation. So, buckle up, and let's get started!

Understanding the Square Root Property

First off, what's the square root property? Simply put, it states that if you have an equation in the form $x^2 = k$, then the solutions for x are $x = \sqrt{k}$ and $x = -\sqrt{k}$. Basically, you take the square root of both sides, remembering that you have both a positive and a negative root. This property stems from the fact that both a positive number and its negative counterpart, when squared, result in a positive number. For instance, both $3^2$ and $(-3)^2$ equal 9.

Now, why is this method so useful? Well, it’s particularly effective when dealing with quadratic equations that lack a linear term (i.e., a term with just x). In these cases, isolating the squared term is straightforward, making the square root property a quick and efficient way to find the solutions. Think about it – trying to factorize $3x^2 - 7 = 0$ directly might seem a bit tricky, but with the square root property, it becomes a breeze. This method is a cornerstone in algebra, offering a direct route to solutions without the need for more complex techniques like factoring or using the quadratic formula in certain scenarios. This foundational understanding not only simplifies equation-solving but also enhances your grasp of mathematical principles, laying a solid groundwork for more advanced algebraic concepts. The square root property showcases the elegance of mathematical tools in simplifying problems, providing a clear and concise path to the solutions. It’s like having a specialized wrench for a specific bolt – it just fits perfectly and gets the job done efficiently!

Step-by-Step Solution for $3x^2 - 7 = 0$

Okay, let's break down the solution to $3x^2 - 7 = 0$ step by step.

1. Isolate the Squared Term: Our mission is to get the $x^2$ term all by itself on one side of the equation. Right now, we've got $3x^2 - 7 = 0$. The first thing we'll do is get rid of that -7. How? By adding 7 to both sides of the equation. This keeps the equation balanced, which is super important. So, we get:

   $3x^2 - 7 + 7 = 0 + 7$

   Which simplifies to:

   $3x^2 = 7$

2. Divide to Get $x^2$ Alone: Now, we've got $3x^2 = 7$, but we want just $x^2$ on the left side. To do that, we need to get rid of the 3 that's multiplying the $x^2$. We can do this by dividing both sides of the equation by 3. Remember, whatever you do to one side, you have to do to the other to keep things fair and square.

   So, we divide both sides by 3:

   $\frac{3x^2}{3} = \frac{7}{3}$

   This simplifies to:

   $x^2 = \frac{7}{3}$

3. Apply the Square Root Property: Alright, we've reached the crucial part where we use the square root property. We've got $x^2$ isolated, so now we take the square root of both sides. And remember, we need to consider both the positive and negative square roots. Think of it like this: both a positive and a negative number, when squared, will give us a positive result. So, we need to account for both possibilities.

   Taking the square root of both sides gives us:

   $x = \pm \sqrt{\frac{7}{3}}$

   The $\pm$ symbol is just a shorthand way of saying "plus or minus," meaning we have two possible solutions: a positive square root and a negative square root. This step is the heart of the square root property – it’s where we actually solve for x by undoing the square.

4. Rationalize the Denominator: We're almost there, but in math, we like to keep things neat and tidy. One of the ways we do this is by rationalizing the denominator. This means we don't want any square roots in the bottom of a fraction. Right now, we have $x = \pm \sqrt{\frac{7}{3}}$. Notice that the square root applies to both the 7 and the 3. So, we can rewrite this as:

   $x = \pm \frac{\sqrt{7}}{\sqrt{3}}$

   To get rid of the $\sqrt{3}$ in the denominator, we multiply both the numerator and the denominator by $\sqrt{3}$. This is like multiplying by 1, so it doesn't change the value of the fraction, just its appearance.

   So, we multiply:

   $x = \pm \frac{\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

   This gives us:

   $x = \pm \frac{\sqrt{7} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}$

   Which simplifies to:

   $x = \pm \frac{\sqrt{21}}{3}$

See how we got rid of the square root in the denominator? That's rationalizing at its finest!

Each of these steps is crucial, guys. Isolating the squared term sets the stage for applying the square root property, which directly gives us the possible values of x. Rationalizing the denominator is the final touch, ensuring our answer is in its simplest, most conventional form. By mastering this process, you'll tackle similar equations with confidence and precision. Remember, each step is a building block, and together, they lead us to the solution. Keep practicing, and you'll become a pro at solving equations using the square root property!

Final Answer

Alright, let's wrap this up! We've gone through the steps, and now we need to present our final answer. Remember, we found that:

$x = \pm \frac{\sqrt{21}}{3}$

This means we have two solutions:

$x = \frac{\sqrt{21}}{3}$ and $x = -\frac{\sqrt{21}}{3}$

So, to write this in the solution set format, we put these values inside curly braces, separated by a comma. The solution set is:

$\left\{ -\frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3} \right\}$

And there you have it! We've successfully solved the equation $3x^2 - 7 = 0$ using the square root property. We isolated the squared term, took the square root of both sides (remembering the plus or minus!), rationalized the denominator, and presented our final answer in the correct format. You did it!

To recap, the final solution set, neatly packaged and ready to go, is $\left\{ -\frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3} \right\}$. This journey through solving the equation highlights the power and elegance of algebraic methods. From isolating terms to rationalizing denominators, each step is a testament to the precision and beauty of mathematics. Now, armed with this knowledge, you're ready to tackle similar challenges with confidence and skill. Remember, guys, math isn't just about finding answers; it's about understanding the process and appreciating the logic behind each step. Keep practicing, keep exploring, and you'll find that the world of mathematics is full of exciting discoveries and solutions waiting to be uncovered!

The solution set is $\left\{ -\frac{\sqrt{21}}{3}, \frac{\sqrt{21}}{3} \right\}$.