Solve The Equation:$\sqrt{55-r}=\sqrt{\frac{r}{10}}$

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Introduction

Solving equations involving square roots can be a challenging task, especially when they involve variables. In this article, we will focus on solving the equation 55−r=r10\sqrt{55-r}=\sqrt{\frac{r}{10}}. This equation involves a square root on both sides, and our goal is to isolate the variable rr and find its value.

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation 55−r=r10\sqrt{55-r}=\sqrt{\frac{r}{10}} states that the square root of 55−r55-r is equal to the square root of r10\frac{r}{10}. This means that the value inside the square root on the left-hand side is equal to the value inside the square root on the right-hand side.

Step 1: Square Both Sides

To solve this equation, we can start by squaring both sides. This will eliminate the square roots and give us a simpler equation to work with. When we square both sides, we get:

(55−r)2=(r10)2\left(\sqrt{55-r}\right)^2 = \left(\sqrt{\frac{r}{10}}\right)^2

Using the property of exponents that (am)n=amn(a^m)^n = a^{mn}, we can simplify this to:

55−r=r1055-r = \frac{r}{10}

Step 2: Multiply Both Sides by 10

To get rid of the fraction, we can multiply both sides of the equation by 10. This will give us:

10(55−r)=10(r10)10(55-r) = 10\left(\frac{r}{10}\right)

Using the distributive property, we can simplify this to:

550−10r=r550-10r = r

Step 3: Add 10r to Both Sides

To isolate the variable rr, we can add 10r to both sides of the equation. This will give us:

550=11r550 = 11r

Step 4: Divide Both Sides by 11

Finally, we can divide both sides of the equation by 11 to solve for rr. This will give us:

r=55011r = \frac{550}{11}

Conclusion

In this article, we solved the equation 55−r=r10\sqrt{55-r}=\sqrt{\frac{r}{10}} by squaring both sides, multiplying both sides by 10, adding 10r to both sides, and finally dividing both sides by 11. This gave us the value of rr as 55011\frac{550}{11}. We hope this article has been helpful in understanding how to solve equations involving square roots.

Tips and Tricks

  • When solving equations involving square roots, it's often helpful to square both sides to eliminate the square roots.
  • When multiplying both sides of an equation by a number, make sure to distribute the multiplication to all terms on the side.
  • When adding or subtracting the same value to both sides of an equation, make sure to do so to all terms on the side.

Real-World Applications

Solving equations involving square roots has many real-world applications. For example, in physics, the equation 55−r=r10\sqrt{55-r}=\sqrt{\frac{r}{10}} can be used to model the motion of an object under the influence of gravity. In engineering, the equation can be used to design systems that involve square roots, such as electronic circuits or mechanical systems.

Common Mistakes

  • When solving equations involving square roots, it's easy to make mistakes by not squaring both sides or by not distributing the multiplication correctly.
  • When adding or subtracting the same value to both sides of an equation, it's easy to make mistakes by not doing so to all terms on the side.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with practice and patience, it can be mastered. By following the steps outlined in this article, you should be able to solve equations involving square roots with ease. Remember to always square both sides, multiply both sides by 10, add 10r to both sides, and finally divide both sides by 11. With these tips and tricks, you'll be well on your way to becoming a master of solving equations involving square roots.

Introduction

In our previous article, we solved the equation 55−r=r10\sqrt{55-r}=\sqrt{\frac{r}{10}} by squaring both sides, multiplying both sides by 10, adding 10r to both sides, and finally dividing both sides by 11. In this article, we will answer some common questions that people may have when solving equations involving square roots.

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

A: The first step in solving an equation involving square roots is to square both sides of the equation. This will eliminate the square roots and give us a simpler equation to work with.

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

A: We need to square both sides of the equation because the square root operation is not commutative, meaning that the order of the terms inside the square root matters. By squaring both sides, we can eliminate the square roots and work with a simpler equation.

Q: What happens if we don't square both sides of the equation?

A: If we don't square both sides of the equation, we will be left with an equation that involves square roots, which can be difficult to solve. Squaring both sides is a crucial step in solving equations involving square roots.

Q: Can we use other methods to solve equations involving square roots?

A: While squaring both sides is a common method for solving equations involving square roots, there are other methods that can be used in certain situations. For example, we can use the property of square roots that a=b\sqrt{a} = \sqrt{b} if and only if a=ba = b. However, this method is not as general as squaring both sides and may not work for all equations.

Q: How do we know when to multiply both sides of the equation by a number?

A: We know when to multiply both sides of the equation by a number when we need to eliminate a fraction or a square root. For example, if we have an equation that involves a fraction, we can multiply both sides by the denominator to eliminate the fraction.

Q: What happens if we multiply both sides of the equation by a number that is not a factor of the equation?

A: If we multiply both sides of the equation by a number that is not a factor of the equation, we may introduce extraneous solutions or make the equation more complicated. It's essential to choose a number that is a factor of the equation to avoid these problems.

Q: Can we use algebraic manipulations to solve equations involving square roots?

A: Yes, we can use algebraic manipulations to solve equations involving square roots. For example, we can use the distributive property to expand expressions, or we can use the commutative property to rearrange terms.

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

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

  • Not squaring both sides of the equation
  • Not distributing the multiplication correctly
  • Not adding or subtracting the same value to both sides of the equation
  • Introducing extraneous solutions by multiplying both sides by a number that is not a factor of the equation

Q: How can we check our solutions to an equation involving square roots?

A: We can check our solutions to an equation involving square roots by plugging the solution back into the original equation and verifying that it is true. If the solution satisfies the original equation, then it is a valid solution.

Conclusion

Solving equations involving square roots can be a challenging task, but with practice and patience, it can be mastered. By following the steps outlined in this article and avoiding common mistakes, you should be able to solve equations involving square roots with ease. Remember to always square both sides, multiply both sides by 10, add 10r to both sides, and finally divide both sides by 11. With these tips and tricks, you'll be well on your way to becoming a master of solving equations involving square roots.