Solve The Equation. You Will Need To Square Both Sides Of The Equation Twice. X − 16 = X − 4 \sqrt{x-16}=\sqrt{x}-4 X − 16 ​ = X ​ − 4 Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Solution Set Is

by ADMIN 256 views

=====================================================

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will guide you through the process of solving the equation x−16=x−4\sqrt{x-16}=\sqrt{x}-4. We will square both sides of the equation twice to find the solution set.

Step 1: Square Both Sides of the Equation

The first step in solving the equation is to square both sides of the equation. This will help us eliminate the square roots and simplify the equation.

(x−16)2=(x−4)2\left(\sqrt{x-16}\right)^2 = \left(\sqrt{x}-4\right)^2

Expanding the squared expressions, we get:

x−16=x−8x+16x-16 = x - 8\sqrt{x} + 16

Step 2: Simplify the Equation

Now, let's simplify the equation by combining like terms.

x−16=x−8x+16x - 16 = x - 8\sqrt{x} + 16

Subtracting xx from both sides, we get:

−16=−8x-16 = -8\sqrt{x}

Step 3: Isolate the Square Root Term

Next, we need to isolate the square root term. To do this, we can divide both sides of the equation by −8-8.

−16−8=−8x−8\frac{-16}{-8} = \frac{-8\sqrt{x}}{-8}

Simplifying the fractions, we get:

2=x2 = \sqrt{x}

Step 4: Square Both Sides Again

Now that we have isolated the square root term, we can square both sides of the equation again to eliminate the square root.

(2)2=(x)2\left(2\right)^2 = \left(\sqrt{x}\right)^2

Expanding the squared expressions, we get:

4=x4 = x

Step 5: Check the Solution

Before we conclude that x=4x=4 is the solution, we need to check if it satisfies the original equation.

x−16=x−4\sqrt{x-16}=\sqrt{x}-4

Substituting x=4x=4 into the equation, we get:

4−16=4−4\sqrt{4-16}=\sqrt{4}-4

Simplifying the expressions, we get:

−12=−2\sqrt{-12}=-2

Since −12\sqrt{-12} is not a real number, we conclude that x=4x=4 is not a solution to the equation.

Conclusion

In this article, we have guided you through the process of solving the equation x−16=x−4\sqrt{x-16}=\sqrt{x}-4. We have squared both sides of the equation twice to find the solution set. However, we have found that the solution x=4x=4 does not satisfy the original equation. Therefore, we conclude that the equation has no solution.

Final Answer

The final answer is: Nosolution\boxed{No solution}

=============================================================

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will answer some of the most frequently asked questions related to solving equations with square roots.

Q: What is the first step in solving an equation with square roots?

A: The first step in solving an equation with square roots is to square both sides of the equation. This will help us eliminate the square roots and simplify the equation.

Q: Why do we need to square both sides of the equation twice?

A: We need to square both sides of the equation twice to eliminate the square roots completely. Squaring both sides once will only eliminate one square root, leaving the other square root intact.

Q: What if the equation has a negative number under the square root?

A: If the equation has a negative number under the square root, it means that the expression inside the square root is not a real number. In this case, the equation has no solution.

Q: How do I check if the solution satisfies the original equation?

A: To check if the solution satisfies the original equation, substitute the solution into the equation and simplify the expressions. If the resulting expression is true, then the solution is correct.

Q: What if the solution does not satisfy the original equation?

A: If the solution does not satisfy the original equation, then it is not a valid solution. In this case, we need to go back and re-examine our work to find the correct solution.

Q: Can I use other methods to solve equations with square roots?

A: Yes, there are other methods to solve equations with square roots, such as using the conjugate or using algebraic manipulations. However, squaring both sides of the equation is a common and efficient method to solve these types of equations.

Q: What are some common mistakes to avoid when solving equations with square roots?

A: Some common mistakes to avoid when solving equations with square roots include:

  • Squaring both sides of the equation only once
  • Not checking if the solution satisfies the original equation
  • Not considering the possibility of a negative number under the square root
  • Not using the correct algebraic manipulations to simplify the equation

Q: How can I practice solving equations with square roots?

A: You can practice solving equations with square roots by working through example problems and exercises. You can also try solving equations with square roots on your own and then check your work to see if you made any mistakes.

Additional Resources

Final Answer

The final answer is: Nosolution\boxed{No solution}