Use the square root property

Oct 13, 2025 by miodeya.com 144 views

Introduction

Equations involving square roots can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $\sqrt{6y + 9} = \sqrt{8y - 71}$ , which involves square roots and algebraic manipulation. We will break down the solution into manageable steps, making it easy to follow and understand.

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 make the equation easier to work with.

$\sqrt{6y + 9} = \sqrt{8y - 71}$

Squaring both sides of the equation gives us:

$6y + 9 = 8y - 71$

Step 2: Simplify the Equation

Now that we have squared both sides of the equation, we can simplify it by combining like terms.

$6y + 9 = 8y - 71$

Subtracting $6y$ from both sides of the equation gives us:

$9 = 2y - 71$

Adding $71$ to both sides of the equation gives us:

$80 = 2y$

Step 3: Solve for y

Now that we have simplified the equation, we can solve for $y$ .

$80 = 2y$

Dividing both sides of the equation by $2$ gives us:

$y = 40$

Conclusion

In this article, we have solved the equation $\sqrt{6y + 9} = \sqrt{8y - 71}$ using a step-by-step approach. We started by squaring both sides of the equation, then simplified it by combining like terms, and finally solved for $y$ . The solution to the equation is $y = 40$ .

Why is this Equation Important?

This equation may seem simple, but it is an important concept in mathematics, particularly in algebra and geometry. Understanding how to solve equations involving square roots is crucial in many real-world applications, such as physics, engineering, and computer science.

Real-World Applications

Equations involving square roots have many real-world applications, including:

Physics: In physics, equations involving square roots are used to describe the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
Engineering: In engineering, equations involving square roots are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Computer Science: In computer science, equations involving square roots are used in algorithms and data structures, such as binary search trees and hash tables.

Tips and Tricks

Here are some tips and tricks to help you solve equations involving square roots:

Use the square root property: The square root property states that if $a^2 = b^2$ , then $a = b$ or $a = -b$ .
Simplify the equation: Simplifying the equation by combining like terms can make it easier to solve.
Check your work: Always check your work by plugging the solution back into the original equation.

Conclusion

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 help eliminate the square roots and make the equation easier 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 a one-to-one function. This means that if $a = b$ , it does not necessarily mean that $\sqrt{a} = \sqrt{b}$ . By squaring both sides of the equation, we can eliminate the square roots and make the equation easier to work with.

Q: How do I simplify an equation involving square roots?

A: To simplify an equation involving square roots, you can combine like terms and eliminate any square roots that are present. For example, if you have the equation $\sqrt{6y + 9} = \sqrt{8y - 71}$ , you can simplify it by combining like terms and eliminating the square roots.

Q: What is the square root property?

A: The square root property states that if $a^2 = b^2$ , then $a = b$ or $a = -b$ . This property is useful when solving equations involving square roots, as it allows us to eliminate the square roots and solve for the variable.

Q: How do I check my work when solving an equation involving square roots?

A: To check your work when solving an equation involving square roots, you can plug the solution back into the original equation and verify that it is true. For example, if you have the equation $\sqrt{6y + 9} = \sqrt{8y - 71}$ and you find that $y = 40$ is a solution, you can plug $y = 40$ back into the original equation and verify that it is true.

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: Failing to square both sides of the equation can lead to incorrect solutions.
Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
Not checking your work: Failing to check your work can lead to incorrect solutions.

Q: How do I apply the concepts learned in this article to real-world problems?

A: The concepts learned in this article can be applied to a wide range of real-world problems, including:

Physics: Equations involving square roots are used to describe the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
Engineering: Equations involving square roots are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Computer Science: Equations involving square roots are used in algorithms and data structures, such as binary search trees and hash tables.

Q: What are some additional resources for learning more about solving equations involving square roots?

A: Some additional resources for learning more about solving equations involving square roots include:

Textbooks: There are many textbooks available that cover the topic of solving equations involving square roots.
Online resources: There are many online resources available that provide step-by-step instructions and examples for solving equations involving square roots.
Practice problems: Practicing solving equations involving square roots is an excellent way to build your skills and confidence.

Conclusion

In conclusion, solving equations involving square roots requires a step-by-step approach, including squaring both sides of the equation, simplifying it by combining like terms, and solving for the variable. By following the tips and tricks outlined in this article, you can become proficient in solving equations involving square roots and apply the concepts learned to real-world problems.