Solving Radical Equations: A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving a radical equation. Radical equations might seem intimidating at first, but don't worry, we'll break it down step by step to make it super easy. We'll tackle the equation $4 + \sqrt{5x + 66} = x + 10$. So, grab your pencils, and let's get started!

Isolating the Radical Term

Alright, so the first thing we need to do when solving radical equations is to isolate the radical term. This means we want to get the square root part all by itself on one side of the equation. In our equation, $4 + \sqrt{5x + 66} = x + 10$, we have $\sqrt{5x + 66}$ hanging out with a '+ 4'. To get rid of that '+ 4', we need to subtract 4 from both sides of the equation. Remember, whatever you do to one side, you gotta do to the other to keep things balanced!

So, let's do it: $4 + \sqrt{5x + 66} - 4 = x + 10 - 4$. Simplifying this gives us $\sqrt{5x + 66} = x + 6$. Awesome! Now we have the radical term all by itself on the left side. This is a crucial step because it sets us up for the next part, which is getting rid of that pesky square root. You might be wondering why we do this first. Well, if we tried to square both sides of the original equation without isolating the radical, we'd end up with a much more complicated mess. Isolating the radical term simplifies the equation and makes it easier to solve. It's like prepping your ingredients before you start cooking – it just makes the whole process smoother! Think of isolating the radical as setting the stage for the main event. Once it's isolated, you can confidently move on to the next step, knowing you've laid a solid foundation. It reduces the chances of making mistakes and keeps everything nice and organized. Trust me; this little trick will save you a lot of headaches down the road! Once the radical is isolated, the next step of squaring both sides is much easier to handle. It's like having a clear path to the solution, making the whole process less intimidating and more manageable. And, of course, keeping the equation balanced is key. Subtracting 4 from both sides ensures that we're not changing the equation's fundamental truth. We're just rearranging things to make it easier to solve.

Squaring Both Sides

Now that we've got the radical isolated, it's time to eliminate the square root. The way we do that is by squaring both sides of the equation. Remember that $\sqrt{a}^2 = a$, so squaring a square root just cancels it out. In our case, we have $\sqrt{5x + 66} = x + 6$. We're going to square both sides like this: $(\sqrt{5x + 66})^2 = (x + 6)^2$.

On the left side, the square root and the square cancel each other out, leaving us with $5x + 66$. On the right side, we need to expand $(x + 6)^2$. Remember that $(x + 6)^2$ means $(x + 6)(x + 6)$. To expand this, we can use the FOIL method (First, Outer, Inner, Last): $(x + 6)(x + 6) = x*x + x*6 + 6*x + 6*6 = x^2 + 6x + 6x + 36 = x^2 + 12x + 36$. So, our equation now looks like this: $5x + 66 = x^2 + 12x + 36$. Squaring both sides might seem like a simple step, but it's a powerful tool in solving radical equations. It allows us to get rid of the square root and transform the equation into a more manageable form. Just remember to square both sides completely and accurately to avoid any errors. After squaring both sides, we're left with a quadratic equation, which we can then solve using various methods like factoring, completing the square, or the quadratic formula. It's like unlocking a new level in the equation-solving game! So, embrace the power of squaring, and watch those square roots disappear! It opens up a whole new world of possibilities for solving equations. Plus, it's kind of satisfying to see that square root vanish, isn't it? It's like saying, "I conquered you, square root!" But remember, with great power comes great responsibility. Make sure you're squaring both sides correctly and paying attention to the details. The FOIL method is your friend here. So, use it wisely and conquer those radical equations like a champ! The end result should be a clean and clear quadratic equation that's ready to be solved.

Rearranging into a Quadratic Equation

Now, we need to rearrange our equation into the standard quadratic form: $ax^2 + bx + c = 0$. We have $5x + 66 = x^2 + 12x + 36$. To get everything on one side, let's subtract $5x$ and $66$ from both sides: $5x + 66 - 5x - 66 = x^2 + 12x + 36 - 5x - 66$. Simplifying this gives us $0 = x^2 + 7x - 30$. Now we have a quadratic equation in the form $x^2 + 7x - 30 = 0$. This form is super useful because it allows us to use different techniques to solve for $x$, such as factoring or using the quadratic formula. Getting the equation into this standard form is like organizing your tools before starting a project. It makes everything easier to handle and ensures you're ready for the next steps. A well-organized quadratic equation is a happy quadratic equation! So, take your time and make sure you're rearranging everything correctly. Double-check your work to avoid any mistakes, and get ready to solve for those x values. The key to rearranging is to make sure the equation is set to zero. That way, you are setting the stage for applying the standard quadratic methods. By setting the equation to zero, we are basically saying that we are looking for the values of x that make the entire equation equal to zero. Those values are what we call the solutions or roots of the quadratic equation. Once we have the equation in standard form, we can easily identify the coefficients a, b, and c, which are essential for using the quadratic formula. So, take a deep breath, rearrange those terms, and get ready to conquer the quadratic equation! You've got this!

Factoring the Quadratic Equation

Okay, now that we have $x^2 + 7x - 30 = 0$, let's try to factor it. We're looking for two numbers that multiply to -30 and add up to 7. Those numbers are 10 and -3, because $10 * -3 = -30$ and $10 + (-3) = 7$. So, we can rewrite the quadratic equation as $(x + 10)(x - 3) = 0$. Now, using the zero-product property, which states that if $a * b = 0$, then either $a = 0$ or $b = 0$, we set each factor equal to zero: $x + 10 = 0$ or $x - 3 = 0$. Solving for $x$ in each case gives us $x = -10$ or $x = 3$. Factoring is a fantastic way to solve quadratic equations because it's often quicker and simpler than other methods. It's like finding the perfect pieces of a puzzle and fitting them together to reveal the solution. When factoring, it's essential to look for the right combination of numbers that satisfy both the multiplication and addition conditions. Once you find those numbers, the rest is smooth sailing! If you have trouble factoring, don't worry! There are plenty of resources available to help you practice and improve your factoring skills. And if factoring just isn't your thing, you can always use the quadratic formula instead. The zero-product property is a powerful tool that allows us to find the solutions once we have factored the quadratic equation. It's like a magic wand that transforms factors into solutions! So, embrace the power of factoring and watch those quadratic equations crumble before you. With a little practice, you'll become a factoring master in no time! It's all about finding the right combination of numbers and using the zero-product property to unlock the solutions. So, keep practicing, and you'll be amazed at how quickly you can factor quadratic equations. And remember, even if you get stuck, there are always other methods you can use to solve for x.

Checking for Extraneous Solutions

Here's a super important step: checking for extraneous solutions. Because we squared both sides of the equation, we might have introduced solutions that don't actually work in the original equation. We need to plug each of our potential solutions back into the original equation, $4 + \sqrt{5x + 66} = x + 10$, to see if they make the equation true.

Let's start with $x = -10$: $4 + \sqrt{5*(-10) + 66} = -10 + 10$. Simplifying this gives us $4 + \sqrt{-50 + 66} = 0$, which is $4 + \sqrt{16} = 0$, so $4 + 4 = 0$, which is $8 = 0$. This is not true, so $x = -10$ is an extraneous solution.

Now let's check $x = 3$: $4 + \sqrt{5*3 + 66} = 3 + 10$. Simplifying this gives us $4 + \sqrt{15 + 66} = 13$, which is $4 + \sqrt{81} = 13$, so $4 + 9 = 13$, which is $13 = 13$. This is true, so $x = 3$ is a valid solution.

Therefore, the only solution to the equation $4 + \sqrt{5x + 66} = x + 10$ is $x = 3$.

Checking for extraneous solutions is like double-checking your work to make sure everything is correct. It's a crucial step that can save you from making mistakes and ensuring you have the right answer. Squaring both sides can introduce solutions that don't actually satisfy the original equation, so it's essential to plug each potential solution back in and see if it works. If a solution doesn't work, it's called an extraneous solution, and we discard it. When checking for extraneous solutions, pay close attention to the order of operations and make sure you're simplifying everything correctly. It's easy to make a mistake, so take your time and double-check your work. Extraneous solutions can be tricky, so it's essential to be thorough and careful when checking your work. And remember, even if a solution seems correct, it's always a good idea to plug it back into the original equation just to be sure. Checking for extraneous solutions is a skill that will serve you well in all areas of mathematics. So, embrace the process and become a master of verification! You'll be glad you took the time to check your work and avoid any potential mistakes.

Conclusion

And there you have it! We successfully solved the radical equation $4 + \sqrt{5x + 66} = x + 10$ and found that the only valid solution is $x = 3$. Remember the key steps: isolate the radical, square both sides, rearrange into a quadratic equation, factor (or use the quadratic formula), and, most importantly, check for extraneous solutions. Keep practicing, and you'll become a pro at solving radical equations in no time! You guys are awesome!