Solving (6x - 5)^2 = 37: A Step-by-Step Guide
Hey guys! Today, we're diving into a fun math problem: solving the equation (6x - 5)^2 = 37 using the square root property. Don't worry, it sounds more intimidating than it actually is. We'll break it down step by step, so you'll be a pro in no time. So, grab your pencils and let's get started!
Understanding the Square Root Property
Before we jump into the problem, let's quickly recap the square root property. In simple terms, it states that if you have an equation in the form of x^2 = a, then x can be either the positive or negative square root of a. Mathematically, we write this as:
x = ±√a
This little property is super handy for solving equations where a squared term is isolated. Now that we've refreshed our memory, let's tackle our equation.
Applying the Square Root Property
In this section, we will apply the square root property to the equation, in order to derive a comprehensive solution for the same. To start, let's look at our equation again:
(6x - 5)^2 = 37
Notice that we have a squared term, (6x - 5)^2, isolated on one side of the equation. This is perfect for using the square root property! The first step is to take the square root of both sides of the equation. Remember, we need to consider both the positive and negative square roots:
√(6x - 5)^2 = ±√37
Taking the square root of (6x - 5)^2 simply gives us 6x - 5. So, our equation now looks like this:
6x - 5 = ±√37
Now we're getting somewhere! We've eliminated the square and have a much simpler equation to solve. The next step involves isolating x, which we'll cover in the next section.
Isolating x
Okay, so we're at 6x - 5 = ±√37. Our mission now is to get x all by itself on one side of the equation. The first thing we need to do is get rid of that -5. We can do this by adding 5 to both sides of the equation:
6x - 5 + 5 = 5 ± √37
This simplifies to:
6x = 5 ± √37
Great! We're one step closer. Now, we need to get rid of the 6 that's multiplying x. To do this, we'll divide both sides of the equation by 6:
6x / 6 = (5 ± √37) / 6
This gives us our solutions for x:
x = (5 ± √37) / 6
So, we actually have two solutions here, one with the plus sign and one with the minus sign. Let's separate them out to make it clearer.
Finding the Two Solutions
As we found in the previous section, we have two possible solutions for x:
x = (5 + √37) / 6
and
x = (5 - √37) / 6
These are the exact solutions. If you need decimal approximations, you can use a calculator to find the square root of 37 (which is approximately 6.08) and then do the arithmetic.
Let's calculate the approximate values:
For x = (5 + √37) / 6:
x ≈ (5 + 6.08) / 6 ≈ 11.08 / 6 ≈ 1.85
For x = (5 - √37) / 6:
x ≈ (5 - 6.08) / 6 ≈ -1.08 / 6 ≈ -0.18
So, our approximate solutions are x ≈ 1.85 and x ≈ -0.18. It's always a good idea to check these solutions by plugging them back into the original equation to make sure they work. This ensures that our calculations are correct and we haven't made any mistakes along the way.
Verifying the Solutions
To verify our solutions, we'll plug each approximate value of x back into the original equation, (6x - 5)^2 = 37, and see if the equation holds true.
Let's start with x ≈ 1.85:
(6 * 1.85 - 5)^2 = (11.1 - 5)^2 = (6.1)^2 = 37.21
This is quite close to 37, so x ≈ 1.85 is a good approximate solution.
Now, let's check x ≈ -0.18:
(6 * -0.18 - 5)^2 = (-1.08 - 5)^2 = (-6.08)^2 = 36.9664
This is also very close to 37, so x ≈ -0.18 is another good approximate solution.
Since our approximations result in values close to 37 when plugged back into the original equation, we can be confident that our solutions are correct. This verification step is crucial in mathematics to ensure the accuracy of our answers.
Common Mistakes to Avoid
When solving equations using the square root property, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you arrive at the correct solution. Let's discuss some of these common errors:
Forgetting the ± Sign
One of the most frequent mistakes is forgetting to include both the positive and negative square roots. When you take the square root of both sides of an equation, remember that there are two possible solutions: one positive and one negative. For example, if you have x^2 = 9, the solutions are x = 3 and x = -3. Neglecting the negative root will lead to an incomplete solution set.
Incorrectly Applying the Square Root Property
Another common mistake is applying the square root property prematurely or incorrectly. The property should only be applied when the squared term is isolated on one side of the equation. For instance, if you have an equation like (x + 2)^2 - 5 = 0, you need to isolate the squared term first by adding 5 to both sides, resulting in (x + 2)^2 = 5. Only then can you apply the square root property.
Arithmetic Errors
Simple arithmetic errors can also lead to incorrect solutions. Be careful when performing calculations, especially when dealing with square roots and fractions. Double-check your work to ensure you haven't made any mistakes in addition, subtraction, multiplication, or division. Using a calculator for complex calculations can help reduce the likelihood of errors.
Not Verifying Solutions
Failing to verify your solutions is another common mistake. Always plug your solutions back into the original equation to check if they are correct. This step is crucial for catching errors and ensuring the accuracy of your answers. If a solution does not satisfy the original equation, it is not a valid solution.
Misunderstanding the Order of Operations
Misunderstanding the order of operations can also lead to errors. Remember to follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions. For example, in the equation (6x - 5)^2 = 37, make sure to perform the operations inside the parentheses first before squaring.
By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving equations using the square root property.
Conclusion
And there you have it! We've successfully solved the equation (6x - 5)^2 = 37 using the square root property. We walked through each step, from understanding the property itself to isolating x and finding our two solutions. Remember, math can be challenging, but with practice and a clear understanding of the steps, you can conquer any equation. Keep practicing, and you'll become a math whiz in no time! If you guys have any questions, feel free to ask. Happy solving!
