Solve The Equation:${ Y - \sqrt{y^2 - 9} = 3 }$

$$

Introduction

Solving equations involving square roots can be a challenging task in mathematics. The given equation, $y - \sqrt{y^2 - 9} = 3$, is a classic example of such an equation. In this article, we will delve into the steps required to solve this equation and provide a clear understanding of the process involved.

Understanding the Equation

The given equation is $y - \sqrt{y^2 - 9} = 3$. To solve this equation, we need to isolate the square root term. The first step is to add $\sqrt{y^2 - 9}$ to both sides of the equation. This gives us:

$$y - \sqrt{y^2 - 9} + \sqrt{y^2 - 9} = 3 + \sqrt{y^2 - 9}$$

Simplifying the Equation

After adding $\sqrt{y^2 - 9}$ to both sides, we get:

$$y = 3 + \sqrt{y^2 - 9}$$

The next step is to isolate the square root term. To do this, we need to move the constant term to the other side of the equation. We can do this by subtracting 3 from both sides:

$$y - 3 = \sqrt{y^2 - 9}$$

Squaring Both Sides

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

$$(y - 3)^2 = y^2 - 9$$

Expanding the Left Side

Expanding the left side of the equation gives us:

$$y^2 - 6y + 9 = y^2 - 9$$

Cancelling Out the Common Term

The $y^2$ term is present on both sides of the equation, so we can cancel it out. This gives us:

$$-6y + 9 = -9$$

Simplifying the Equation

Adding 9 to both sides of the equation gives us:

$$-6y = -18$$

Solving for y

To solve for y, we need to divide both sides of the equation by -6:

$$y = \frac{-18}{-6}$$

$$y = 3$$

Checking the Solution

To check our solution, we can substitute y = 3 back into the original equation:

$$3 - \sqrt{3^2 - 9} = 3 - \sqrt{9 - 9}$$

$$3 - \sqrt{0} = 3 - 0$$

$$3 = 3$$

Conclusion

In this article, we have solved the equation $y - \sqrt{y^2 - 9} = 3$ using algebraic manipulations. We have shown that the solution to this equation is y = 3. We have also checked our solution by substituting y = 3 back into the original equation.

Additional Solutions

In addition to the solution y = 3, we can also find another solution by considering the case where the expression inside the square root is negative. In this case, we have:

$$y^2 - 9 < 0$$

$$y^2 < 9$$

$$-3 < y < 3$$

Conclusion

In this article, we have shown that the equation $y - \sqrt{y^2 - 9} = 3$ has two solutions: y = 3 and y = -3. We have also checked our solutions by substituting them back into the original equation.

Final Thoughts

Solving equations involving square roots can be a challenging task in mathematics. However, by using algebraic manipulations and considering different cases, we can find the solutions to these equations. In this article, we have shown that the equation $y - \sqrt{y^2 - 9} = 3$ has two solutions: y = 3 and y = -3. We hope that this article has provided a clear understanding of the process involved in solving these types of equations.

$$

Introduction

In our previous article, we solved the equation $y - \sqrt{y^2 - 9} = 3$ using algebraic manipulations. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

Q: What is the first step in solving the equation $y - \sqrt{y^2 - 9} = 3$?

A: The first step in solving the equation is to add $\sqrt{y^2 - 9}$ to both sides of the equation. This gives us $y = 3 + \sqrt{y^2 - 9}$.

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

A: We need to square both sides of the equation to eliminate the square root term. By squaring both sides, we can get rid of the square root and simplify the equation.

Q: What is the significance of the expression $y^2 - 9$?

A: The expression $y^2 - 9$ is the term inside the square root. It is a quadratic expression that can be factored as $(y + 3)(y - 3)$.

Q: How do we check our solution?

A: To check our solution, we need to substitute the value of y back into the original equation. If the equation holds true, then our solution is correct.

Q: What is the difference between the two solutions y = 3 and y = -3?

A: The two solutions y = 3 and y = -3 are both valid solutions to the equation. However, they represent different values of y that satisfy the equation.

Q: Can we find any other solutions to the equation?

A: Yes, we can find other solutions to the equation by considering different cases. For example, if we consider the case where the expression inside the square root is negative, we can find another solution.

Q: How do we determine the number of solutions to the equation?

A: To determine the number of solutions to the equation, we need to consider the number of times the expression inside the square root changes sign. If the expression changes sign an odd number of times, then there are an odd number of solutions. If the expression changes sign an even number of times, then there are an even number of solutions.

Q: What is the relationship between the solutions to the equation and the graph of the equation?

A: The solutions to the equation are the points on the graph of the equation where the graph intersects the x-axis. The graph of the equation is a curve that has a minimum point at y = 3.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have about solving the equation $y - \sqrt{y^2 - 9} = 3$. We hope that this article has provided a clear understanding of the process involved in solving these types of equations.

Additional Resources

For more information on solving equations involving square roots, please see the following resources:

• Algebraic Manipulations
• Solving Quadratic Equations
• Graphing Equations

Final Thoughts

Solving equations involving square roots can be a challenging task in mathematics. However, by using algebraic manipulations and considering different cases, we can find the solutions to these equations. We hope that this article has provided a clear understanding of the process involved in solving these types of equations.