Type The Correct Answer In The Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar.What Is The Solution To This Equation? ( 1 8 − X ) = − 1 2 \sqrt{\left(\frac{1}{8}-x\right)}=-\frac{1}{2} ( 8 1 ​ − X ) ​ = − 2 1 ​ The Solution Is □ \square □

Understanding the Equation

The given equation is $\sqrt{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}$. To solve this equation, we need to isolate the variable $x$. The equation involves a square root, which can be eliminated by squaring both sides of the equation.

Step 1: Square Both Sides of the Equation

To eliminate the square root, we square both sides of the equation. This gives us:

$\left(\sqrt{\left(\frac{1}{8}-x\right)}\right)^2 = \left(-\frac{1}{2}\right)^2$

Step 2: Simplify the Equation

Squaring both sides of the equation simplifies it to:

$\frac{1}{8}-x = \frac{1}{4}$

Step 3: Isolate the Variable $x$

To isolate the variable $x$, we need to get rid of the fraction $\frac{1}{8}$. We can do this by subtracting $\frac{1}{8}$ from both sides of the equation:

$-x = \frac{1}{4} - \frac{1}{8}$

Step 4: Simplify the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we need to find a common denominator. The least common multiple of $4$ and $8$ is $8$. So, we can rewrite the equation as:

$-x = \frac{2}{8} - \frac{1}{8}$

Step 5: Simplify the Right-Hand Side of the Equation

Simplifying the right-hand side of the equation gives us:

$-x = \frac{1}{8}$

Step 6: Multiply Both Sides of the Equation by $-1$

To isolate the variable $x$, we need to multiply both sides of the equation by $-1$:

$x = -\frac{1}{8}$

Step 7: Write the Final Answer

The final answer is $\boxed{-\frac{1}{8}}$.

Conclusion

In this article, we solved the equation $\sqrt{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}$. We eliminated the square root by squaring both sides of the equation and then isolated the variable $x$ by subtracting $\frac{1}{8}$ from both sides of the equation. The final answer is $\boxed{-\frac{1}{8}}$.

Frequently Asked Questions

• Q: What is the solution to the equation $\sqrt{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}$? A: The solution to the equation is $\boxed{-\frac{1}{8}}$.
• Q: How do I eliminate the square root in the equation? A: You can eliminate the square root by squaring both sides of the equation.
• Q: How do I isolate the variable $x$ in the equation? A: You can isolate the variable $x$ by subtracting $\frac{1}{8}$ from both sides of the equation.

Related Articles

• Solving Equations with Square Roots
• Isolating Variables in Algebraic Equations
• Simplifying Fractions in Algebraic Equations
  

Q&A: Solving Equations with Square Roots

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

A: The first step in solving an equation with a square root is to eliminate the square root by squaring both sides of the equation.

Q: How do I eliminate the square root in an equation?

A: You can eliminate the square root by squaring both sides of the equation. This is because the square root and the square cancel each other out.

Q: What is the next step after eliminating the square root?

A: After eliminating the square root, you need to simplify the equation and isolate the variable.

Q: How do I simplify the equation after eliminating the square root?

A: You can simplify the equation by combining like terms and performing any necessary operations to isolate the variable.

Q: What is the final step in solving an equation with a square root?

A: The final step in solving an equation with a square root is to check your solution by plugging it back into the original equation.

Q: Why is it important to check your solution?

A: It is important to check your solution to ensure that it is correct and that it satisfies the original equation.

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:

• Not eliminating the square root before simplifying the equation
• Not checking the solution after finding it
• Not considering the possibility of multiple solutions

Q: How do I know if I have found the correct solution?

A: You can check if you have found the correct solution by plugging it back into the original equation and verifying that it satisfies the equation.

Q: What if I have multiple solutions to the equation?

A: If you have multiple solutions to the equation, you need to check each solution to see if it satisfies the original equation.

Q: How do I know if a solution is extraneous?

A: A solution is extraneous if it does not satisfy the original equation.

Q: What if I am unsure about how to solve an equation with a square root?

A: If you are unsure about how to solve an equation with a square root, you can try:

• Breaking down the equation into smaller parts
• Using algebraic properties to simplify the equation
• Checking your work and looking for mistakes

Q: Where can I find more information about solving equations with square roots?

A: You can find more information about solving equations with square roots in algebra textbooks, online resources, and math websites.

Conclusion

Solving equations with square roots can be challenging, but with practice and patience, you can become proficient in solving these types of equations. Remember to eliminate the square root, simplify the equation, and check your solution to ensure that it is correct.

Related Articles

• Solving Equations with Square Roots
• Isolating Variables in Algebraic Equations
• Simplifying Fractions in Algebraic Equations
• Algebraic Properties and Operations
• Checking Solutions in Algebraic Equations

Additional Resources

• Khan Academy: Solving Equations with Square Roots
• Mathway: Solving Equations with Square Roots
• Algebra.com: Solving Equations with Square Roots
• Purplemath: Solving Equations with Square Roots