Master the Square Root Property: Solving X²-8x+16=9+16 Equation
Now that you have encountered the equation X² - 8x + 16 = 9 + 16, it's time to apply the square root property to solve it. This equation may seem complicated at first glance, but with the right approach, you can easily find its solution. As we delve deeper into this topic, we will explore the ins and outs of the square root property, and how it can be used to solve quadratic equations.
Before we begin, it's important to understand what a quadratic equation is. A quadratic equation is an equation in which one of the variables is squared, and the highest degree of the variable is two. In our case, X² - 8x + 16 = 9 + 16 is a quadratic equation because the variable X is squared.
The first step in solving this equation using the square root property is to isolate the variable on one side of the equation. To do this, we must move all the constants to the other side of the equation, and then factor the left-hand side. By doing this, we can rewrite the equation as (X-4)²=25.
Now that we have our equation in this form, we can apply the square root property. The square root property states that if (X-4)²=25, then X-4=±5. By adding 4 to both sides of the equation, we get X=4±5.
However, we're not done yet. We need to check our solutions to ensure they are valid. To do this, we must substitute each value of X back into the original equation to see if it holds true. If it does, then the solution is valid. If not, then we must discard it.
Let's start by substituting X=4+5 into the original equation. We get (4+5)²-8(4+5)+16=9+16, which simplifies to 25=25. This solution is valid.
Now let's substitute X=4-5 into the original equation. We get (4-5)²-8(4-5)+16=9+16, which simplifies to 25=25. This solution is also valid.
Therefore, the solutions to the equation X² - 8x + 16 = 9 + 16 are X=4+5 and X=4-5, which simplify to X=9 and X=-1, respectively.
The square root property is an essential tool for solving quadratic equations, and it can be used in a variety of different contexts. By mastering this technique, you can solve a wide range of problems and gain a deeper understanding of mathematics. So next time you encounter a quadratic equation, don't be intimidated – just remember to apply the square root property, and you'll be on your way to finding the solution!
Understanding the Square Root Property
The square root property is an important tool used in algebra to solve equations that involve a square root. It states that if a squared variable is equal to a constant, then the variable can be isolated by taking the square root of both sides of the equation. This allows us to find the value of the variable that satisfies the equation.
Breaking Down the Equation
Let's take a closer look at the equation X² - 8x + 16 = 9 + 16. To apply the square root property, we need to isolate the squared variable on one side of the equation. First, we can simplify the equation by combining like terms.
Combining Like Terms
X² - 8x + 16 = 25
By subtracting 16 from both sides of the equation, we get:
Subtracting 16 From Both Sides
X² - 8x = 9
The Square Root Property in Action
Now that we have isolated the squared variable, we can apply the square root property. By taking the square root of both sides of the equation, we can solve for X.
Taking the Square Root of Both Sides
√(X² - 8x) = √9
Simplifying the right-hand side gives us:
Simplifying the Right-Hand Side
√(X² - 8x) = 3
To solve for X, we need to square both sides of the equation. This will eliminate the square root on the left-hand side.
Squaring Both Sides of the Equation
(X² - 8x) = 9
Now we can simplify this equation by combining like terms and moving all terms to one side of the equation.
Combining Like Terms and Moving All Terms to One Side
X² - 8x - 9 = 0
Factoring the Quadratic Equation
To solve for X, we can factor the quadratic equation. This will give us two solutions for X that satisfy the equation.
Factoring the Quadratic Equation
(X - 9)(X + 1) = 0
From this equation, we can see that either X - 9 = 0 or X + 1 = 0. Solving for X in each case gives us:
Solving for X
X = 9 or X = -1
Checking Our Solutions
To confirm that these solutions are correct, we can substitute them back into the original equation and see if they satisfy it.
Substituting X = 9
X² - 8x + 16 = 9 + 16
81 - 72 + 16 = 25
25 = 25
The equation is satisfied, so X = 9 is a valid solution.
Substituting X = -1
X² - 8x + 16 = 9 + 16
1 + 8 + 16 = 25
25 = 25
The equation is satisfied, so X = -1 is also a valid solution.
Conclusion
In conclusion, applying the square root property is a powerful tool for solving equations that involve a square root. By isolating the squared variable and taking the square root of both sides, we can find the solutions that satisfy the equation. In this case, we used the square root property to solve the equation X² - 8x + 16 = 9 + 16 and found that X = 9 or X = -1 satisfy the equation. By checking our solutions, we confirmed that they are valid and that the equation has been solved correctly.
Understanding the Square Root Property
As you begin to solve the equation X² - 8x + 16 = 9 + 16, it is important to first gain a clear understanding of the Square Root Property. This property states that if a² = b, then a equals the positive and negative square root of b. In other words, if you have an equation in the form of x² = c, you can take the square root of both sides to solve for x.Simplifying the Left Side of the Equation
To apply the Square Root Property to X² - 8x + 16 = 9 + 16, you must first simplify the left side of the equation. This can be done by factoring the quadratic expression into (x-4)². Therefore, the equation becomes (x-4)² = 25.Simplifying the Right Side of the Equation
In addition to simplifying the left side of the equation, it is also necessary to simplify the right side of the equation before applying the Square Root Property. By combining like terms, we can see that 9 + 16 = 25.Isolating the Variable
To solve for X, the variable must be isolated on one side of the equation. To do so, it is necessary to move all constants and terms without an X to the opposite side of the equation. Therefore, we subtract 25 from both sides of the equation, resulting in (x-4)² - 25 = 0.Applying the Square Root Property
Now that the equation has been simplified and the X has been isolated to one side of the equation, it is time to apply the Square Root Property. Taking the square root of both sides of the equation, we get (x-4) = ±5.Identifying the Two Possible Solutions
It is important to remember that using the Square Root Property will result in two possible solutions for X. In this case, the two possible solutions are x = 9 and x = -1.Evaluating Both Solutions
After identifying the two possible solutions for X, it is necessary to evaluate both of them to determine if they are valid solutions. By plugging x = 9 and x = -1 back into the original equation, we can see that both solutions satisfy the equation.Checking for Extraneous Solutions
It is possible that one or both of the solutions obtained through the Square Root Property are extraneous, meaning that when plugged back into the original equation, they do not satisfy the equation. However, in this case, both solutions are valid and there are no extraneous solutions.Writing the Final Answer
Once you have identified the valid solution(s) and ensured that there are no extraneous solutions, it is time to write the final answer to the equation. Therefore, the final answer is x = 9 and x = -1.Recognizing the Importance of the Square Root Property
Overall, the Square Root Property is an important tool for solving quadratic equations and understanding it can help you effectively solve problems like X² - 8x + 16 = 9 + 16. By following the steps outlined above and practicing with various equations, you can become more comfortable with using the Square Root Property to solve quadratic equations.Now That You Have X² - 8x + 16 = 9 + 16, Apply The Square Root Property To The Equation
The Story Behind X² - 8x + 16 = 9 + 16
Imagine yourself in a math class, trying to solve a complex equation. You have been given the equation X² - 8x + 16 = 9 + 16 and asked to solve it. At first glance, it looks intimidating, but you take a deep breath and dive into it.
You start by simplifying the equation by combining the numbers on the right-hand side of the equation. You end up with X² - 8x + 16 = 25.
Next, you remember the square root property and realize that you can solve for X by taking the square root of both sides of the equation.
Applying the Square Root Property
Here's how you apply the square root property:
- Isolate the squared term on one side of the equation (in this case, X²).
- Take the square root of both sides of the equation.
- Solve for X.
Using this method, you isolate the squared term X² on one side of the equation.
X² - 8x + 16 = 25
X² - 8x - 9 = 0
Next, you take the square root of both sides of the equation:
√(X² - 8x - 9) = ± √0
Simplifying the equation, you get:
X - 4 = ±0
X = 4 ± 0
Finally, you solve for X and get the solution:
X = 4
The Importance of Applying the Square Root Property
Applying the square root property is a crucial step in solving complex equations. It allows you to isolate the variable and solve for it, even in situations where the equation seems daunting.
By breaking down the equation into smaller steps and applying the square root property, you can solve even the most challenging problems with ease.
Table Information
| Keyword | Definition |
|---|---|
| Square Root Property | A method of solving quadratic equations by taking the square root of both sides of the equation. |
| X | The variable in the equation, representing an unknown value. |
| Equation | A mathematical statement that shows the equality of two expressions. |
Closing Message: Apply the Square Root Property to the Equation X² - 8x + 16 = 9 + 16
As we conclude this article, I hope that you have gained a better understanding of how to apply the square root property to solve quadratic equations. We have taken a closer look at the steps involved in solving an equation such as X² - 8x + 16 = 9 + 16, and I trust that you are now more confident in your ability to tackle similar problems.
It is important to remember that solving quadratic equations can be challenging, but with the right approach and a solid understanding of the key concepts, you can successfully find the solutions you need. The square root property is just one of the many tools available to you, and it can be particularly useful when you are dealing with equations that involve perfect squares.
Throughout this article, we have emphasized the importance of taking your time and carefully working through each step of the process. It is also essential to keep in mind that practice is key. The more you work on solving quadratic equations, the more comfortable you will become with the techniques involved, and the more successful you will be in your efforts.
As you move forward, I encourage you to continue exploring different methods for solving quadratic equations. There are many resources available to you, including textbooks, online tutorials, and practice problems. By taking advantage of these resources and working diligently to improve your skills, you can become a master of quadratic equations in no time.
Finally, I want to thank you for taking the time to read this article. I hope that you have found it informative and helpful, and that you are now better equipped to solve quadratic equations using the square root property. If you have any questions or comments, please feel free to reach out to us.
Once again, thank you for visiting our blog, and we wish you all the best in your future studies and endeavors!
Answering People Also Ask About Now That You Have X² - 8x + 16 = 9 + 16, Apply The Square Root Property To The Equation
What is the Square Root Property?
The square root property is a method used to solve quadratic equations. It states that if x² = a, then x equals the positive or negative square root of a.
How do you apply the Square Root Property to the equation X² - 8x + 16 = 9 + 16?
To apply the square root property to the equation X² - 8x + 16 = 9 + 16, we need to first simplify the equation:
- X² - 8x + 16 = 25
- X² - 8x - 9 = 0
Now that we have a quadratic equation in standard form, we can apply the square root property:
- Add 9 to both sides: X² - 8x = 9
- Take the square root of both sides: √(X² - 8x) = ±3
- Solve for x by setting up two equations: X - 4 = 3 and X - 4 = -3
- Solve for x in each equation: X = 7 and X = 1
Therefore, the solutions to the equation X² - 8x + 16 = 9 + 16 are X = 7 and X = 1.
Why is it important to use the Square Root Property?
The square root property is important because it provides a method for solving quadratic equations that cannot be factored. By using the square root property, we can find the solutions to these equations and solve real-world problems involving quadratic relationships.
Conclusion
In conclusion, applying the square root property to the equation X² - 8x + 16 = 9 + 16 involves simplifying the equation, adding or subtracting constants, taking the square root of both sides, and solving for x. The square root property is important because it provides a method for solving quadratic equations, which are important in many areas of math and science.