Discover the Equation Resulting from Applying Square Root to (X-9)² = 81 - A Guide to Solving Algebraic Problems
Find out the equation that results from taking the square root of both sides of (X – 9)2 = 81 with our simple guide.
If you've ever wondered how to solve equations involving square roots, you're in the right place. In this article, we'll explore what happens when you take the square root of both sides of an equation, specifically (X – 9)2 = 81. This simple equation may seem like a breeze, but it's actually a great example to learn from. By taking the square root of both sides, we can uncover the value of X and understand how to approach similar problems in the future.
Before diving into the equation, let's review what taking the square root means. The square root of a number is the value that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3 because 3 x 3 = 9. When we take the square root of both sides of an equation, we're essentially trying to isolate the variable (in this case, X) and find its value.
Now, back to the equation at hand. To take the square root of (X – 9)2 = 81, we need to remember a key rule: the square root of a squared number is simply the absolute value of that number. In other words, √(x2) = |x|. Applying this rule to our equation, we get:
|X – 9| = 9
This new equation may look different, but it's essentially the same as the original. The absolute value bars indicate that the result of X – 9 could be either positive or negative, since both values would give us a product of 81 when squared. Therefore, we need to consider two possibilities:
X – 9 = 9 or X – 9 = -9
If we add 9 to both sides of the first equation, we get X = 18. If we add 9 to both sides of the second equation, we get X = 0. These are our two solutions for X.
Now that we've solved the equation, let's take a step back and think about why this process works. When we take the square root of both sides, we're essentially undoing the squaring operation. By doing so, we're left with the original value of X – 9, which we can then solve for X using simple algebraic steps.
It's worth noting that not all equations involving square roots will be as straightforward as this one. In some cases, we may need to simplify the equation further before we can take the square root. Additionally, there may be cases where the equation has no real solutions (i.e. the square root of a negative number), in which case we would need to use imaginary numbers.
So, why is it important to know how to take the square root of both sides of an equation? Well, for starters, it's a key skill in algebra and can help us solve a wide range of problems. It also helps us understand the fundamental concepts behind square roots and how they relate to other mathematical operations.
In conclusion, taking the square root of both sides of an equation is a powerful tool that can help us solve algebraic problems. By understanding how to apply this rule, we can uncover the values of variables and gain a deeper understanding of mathematical concepts. So, the next time you encounter an equation involving square roots, remember the steps we've covered here and approach it with confidence!
Introduction
Mathematics can be challenging, but it is also fascinating. One of the most exciting aspects of math is the ability to solve equations and find solutions to problems. In this article, we will explore the equation that results from taking the square root of both sides of (x – 9)² = 81.
The Square Root Function
The square root function is one of the fundamental mathematical functions. It is the inverse of the squared function, which means that if you square a number and then take the square root of the result, you get back the original number. For example, the square root of 4 is 2 because 2² = 4. Similarly, the square root of 25 is 5 because 5² = 25.
The Equation (x – 9)² = 81
The equation (x – 9)² = 81 is a quadratic equation. It represents a parabola, which is a symmetrical curve that opens either upwards or downwards. The parabola represented by this equation has its vertex at the point (9,0), and its axis of symmetry is the vertical line x = 9.
Taking the Square Root of Both Sides of the Equation
To solve the equation (x – 9)² = 81 by taking the square root of both sides, we need to isolate the squared term on one side of the equation. We can do this by taking the square root of both sides, which gives us:
√[(x – 9)²] = ±9
x – 9 = ±9
Now we can solve for x by adding 9 to both sides of the equation:
x – 9 + 9 = ±9 + 9
x = ±9 + 9
Therefore, the solutions to the equation (x – 9)² = 81 are x = 18 and x = 0.
Checking the Solutions
To check our solutions, we can substitute them back into the original equation and see if they satisfy it. If a solution satisfies the equation, it is a valid solution; if it does not, it is not a valid solution. Let's check our solutions:
For x = 18:
(18 – 9)² = 81
81 = 81 (satisfied)
For x = 0:
(0 – 9)² = 81
81 = 81 (satisfied)
Therefore, our solutions are valid.
Conclusion
In conclusion, taking the square root of both sides of the equation (x – 9)² = 81 yields the solutions x = 18 and x = 0. These solutions satisfy the original equation and are therefore valid. The square root function is a powerful tool in mathematics that allows us to solve equations and find solutions to problems.
Understanding the Equation
As you start delving into the world of Mathematics, you will come across various equations that require your attention. One such equation is (X – 9)2 = 81. But what does it mean, and how do you solve it?Breaking Down the Equation
To fully comprehend this equation, let's start by breaking it down. (X – 9)2 means to square the quantity of X minus 9. So, when you simplify this expression, you get (X – 9) multiplied by itself.Solving for X
Now, your ultimate goal is to solve for X. This means finding the value of X that makes the equation true. So, how do you go about doing this?Taking the Square Root of Both Sides
One common technique used in Math is taking the square root of both sides of an equation. This helps to simplify some equations and solve for variables easily. In this case, we will be taking the square root of both sides of (X – 9)2 = 81.The Square Root of 81
The square root of 81 is 9. This means that if you take the square root of the right side of the equation, you will get 9.Taking the Square Root of the Left Side
Now, let's take the square root of the left side of the equation. When you take the square root of (X – 9)2, you get X – 9.Simplifying the Equation
To simplify the equation further, you can rewrite it as X - 9 = 9.Solving for X
To find the value of X that satisfies the equation, simply add 9 to both sides of the equation. This results in X = 18.Checking Your Answer
It is always a good practice to check your answer by substituting X with 18 in the original equation. When you do this, you get (18 – 9)2 = 81, which indeed holds true.Conclusion
In conclusion, taking the square root of both sides of (X – 9)2 = 81 results in the equation X - 9 = 9, which can be solved to find X = 18. Mastery of these basic algebraic techniques is essential in solving more complex equations in Mathematics. By understanding the equation, breaking it down, and utilizing common techniques such as taking the square root of both sides, you can solve for variables with ease. Remember to always check your answer to ensure accuracy.The Square Root of Both Sides of (X – 9)2 = 81
Once upon a time, there was a young student named Emily who was struggling with solving equations. One day, her math teacher introduced her to the concept of taking the square root of both sides of an equation. Emily was thrilled to learn this new technique and decided to try it out on the equation (X – 9)2 = 81.
What is Taking the Square Root?
Before we dive into Emily's story, let's first understand what it means to take the square root of a number. The square root of a number is a value that when multiplied by itself gives the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.
Table: Square Roots of Common Numbers
| Number | Square Root |
|---|---|
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
Now back to Emily's story. She was given the equation (X – 9)2 = 81 and was asked to solve for X. Emily remembered that she could take the square root of both sides of the equation to simplify it.
- Step 1: Take the Square Root of Both Sides
Emily knew that taking the square root of both sides would eliminate the exponent of 2 on the left side of the equation. She also knew that taking the square root of 81 would give her two possible values, 9 and -9. Therefore, she wrote:
√(X – 9)2 = ±9
- Step 2: Solve for X
Next, Emily needed to solve for X. To do this, she isolated X by rewriting the equation with only X on one side:
X – 9 = ±9
She then solved for X by adding 9 to both sides:
X = 9 ± 9
Emily simplified the equation further and got:
X = 18 or X = 0
Conclusion
Thanks to her math teacher's guidance, Emily was able to solve the equation (X – 9)2 = 81 by taking the square root of both sides. She learned that taking the square root of an equation can simplify it and make it easier to solve. Emily was proud of herself for learning this new technique and felt more confident in her math skills.
Thank you for reading about taking the square root of both sides of (x-9)^2=81
As we come to the end of this blog post, we hope that you have gained a better understanding of how to take the square root of both sides of an equation. This concept is important in algebra and can be used in many different situations.
If you are solving an equation that involves a perfect square, like (x-9)^2=81, taking the square root of both sides will help you find the value of x. By doing this, we can eliminate the exponent and simplify the equation.
It's important to remember that when you take the square root of both sides of an equation, you must consider both the positive and negative square roots. In the case of (x-9)^2=81, we get two solutions: x=12 and x=-6.
Another thing to keep in mind is that sometimes taking the square root of both sides may not result in a real solution. For example, if we have an equation like x^2=-1, taking the square root of both sides would result in x=i or x=-i, which are imaginary solutions.
Now that we have discussed the basics of taking the square root of both sides of an equation, let's look at some examples to help solidify our understanding.
Example 1: Solve for x in the equation (x-4)^2=16.
We begin by taking the square root of both sides: √(x-4)^2=√16. This simplifies to x-4=±4. Adding 4 to both sides gives us x=8 or x=0.
Example 2: Solve for x in the equation 2x^2-6x+4=0.
First, we need to put the equation in standard form: 2(x-1)^2=2. Then we take the square root of both sides: √2(x-1)^2=√2. This simplifies to x-1=±√(2/2). Adding 1 to both sides gives us x=1±1. We get two solutions: x=2 and x=0.
As you can see from these examples, taking the square root of both sides of an equation can help us solve for unknown variables. It's a useful tool to have in your algebra toolbox!
We hope that this blog post has been helpful in explaining the concept of taking the square root of both sides of an equation. If you have any questions or comments, please feel free to leave them below. We always enjoy hearing from our readers.
Thank you for taking the time to read this post. We hope that you have learned something new today.
People Also Ask About Which Equation Results From Taking The Square Root Of Both Sides Of (X – 9)2 = 81?
What is the equation after taking the square root of both sides of (x-9)^2 = 81?
The equation that results from taking the square root of both sides of (x-9)^2 = 81 is:
x - 9 = ±9
This equation can be simplified to two separate equations:
- x - 9 = 9
- x - 9 = -9
Solving for x in each equation, we get:
- x = 18
- x = 0
Why do we take the square root of both sides of an equation?
We take the square root of both sides of an equation to isolate the variable and solve for its value. By taking the square root, we can eliminate the exponent and simplify the equation, making it easier to solve.
What is the importance of checking solutions obtained from taking the square root of both sides of an equation?
It is important to check solutions obtained from taking the square root of both sides of an equation because sometimes, extraneous solutions may arise. These are solutions that do not satisfy the original equation, and can occur when squaring both sides of an equation. Checking the solutions ensures that we have found the correct values for the variable that satisfy the given equation.