Solve The Equation For { Y $} . . . { \sqrt{y+4}=x+2 \}

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Introduction

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In this article, we will delve into solving a quadratic equation involving a square root. The given equation is $\sqrt{y+4}=x+2$. Our goal is to isolate the variable $y$ and find its value. We will use algebraic manipulations to simplify the equation and solve for $y$.

Understanding the Equation

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The given equation involves a square root, which can be challenging to solve. However, we can start by isolating the square root term on one side of the equation. The equation is $\sqrt{y+4}=x+2$. To isolate the square root term, we can square both sides of the equation.

Squaring Both Sides

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Squaring both sides of the equation gives us:

$$\left(\sqrt{y+4}\right)^2=(x+2)^2$$

This simplifies to:

$$y+4=x^2+4x+4$$

Simplifying the Equation

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Now, we can simplify the equation by subtracting $4$ from both sides:

$$y=x^2+4x$$

Isolating y

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Our goal is to isolate the variable $y$. To do this, we can subtract $x^2$ from both sides of the equation:

$$y-x^2=4x$$

Adding $x^2$ to Both Sides

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Adding $x^2$ to both sides of the equation gives us:

$$y=x^2+4x$$

This is the final form of the equation, and we have successfully isolated the variable $y$.

Conclusion

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In this article, we solved the equation $\sqrt{y+4}=x+2$ for the variable $y$. We used algebraic manipulations to simplify the equation and isolate the variable $y$. The final form of the equation is $y=x^2+4x$. This equation can be used to find the value of $y$ for any given value of $x$.

Example Use Case

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Suppose we want to find the value of $y$ when $x=2$. We can substitute $x=2$ into the equation $y=x^2+4x$:

$$y=(2)^2+4(2)$$

This simplifies to:

$$y=4+8$$

$$y=12$$

Therefore, the value of $y$ when $x=2$ is $12$.

Tips and Tricks

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When solving equations involving square roots, it's essential to isolate the square root term on one side of the equation. This allows us to square both sides of the equation and simplify the equation. Additionally, be careful when subtracting or adding terms to both sides of the equation, as this can affect the final form of the equation.

Frequently Asked Questions

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Q: What is the final form of the equation?

A: The final form of the equation is $y=x^2+4x$.

Q: How do I solve the equation for a specific value of $x$?

A: To solve the equation for a specific value of $x$, substitute the value of $x$ into the equation $y=x^2+4x$ and simplify.

Q: What is the purpose of squaring both sides of the equation?

A: Squaring both sides of the equation allows us to eliminate the square root term and simplify the equation.

References

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• [1] Algebraic Manipulations. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/manipulations.html
• [2] Square Root. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/square-root.html

Note: The references provided are for general information purposes only and are not specific to the equation $\sqrt{y+4}=x+2$.

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Introduction

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In our previous article, we solved the equation $\sqrt{y+4}=x+2$ for the variable $y$. We used algebraic manipulations to simplify the equation and isolate the variable $y$. In this article, we will provide a Q&A section to address common questions and concerns related to solving the equation.

Q&A

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Q: What is the purpose of squaring both sides of the equation?

A: Squaring both sides of the equation allows us to eliminate the square root term and simplify the equation. This is a common technique used in algebra to solve equations involving square roots.

Q: How do I know when to square both sides of the equation?

A: You should square both sides of the equation when you have a square root term on one side of the equation. This will allow you to eliminate the square root term and simplify the equation.

Q: What is the final form of the equation?

A: The final form of the equation is $y=x^2+4x$. This equation can be used to find the value of $y$ for any given value of $x$.

Q: How do I solve the equation for a specific value of $x$?

A: To solve the equation for a specific value of $x$, substitute the value of $x$ into the equation $y=x^2+4x$ and simplify.

Q: What if I have a negative value for $x$?

A: If you have a negative value for $x$, you will need to take the absolute value of $x$ before substituting it into the equation. This is because the equation $y=x^2+4x$ is only defined for non-negative values of $x$.

Q: Can I use this equation to solve for $x$ instead of $y$?

A: Yes, you can use this equation to solve for $x$ instead of $y$. To do this, you will need to rearrange the equation to isolate $x$. This will give you a quadratic equation in terms of $x$.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is an equation in which the highest power of the variable is 2, whereas a linear equation is an equation in which the highest power of the variable is 1.

Q: Can I use this equation to solve for $y$ if I have a quadratic equation in terms of $x$?

A: Yes, you can use this equation to solve for $y$ if you have a quadratic equation in terms of $x$. To do this, you will need to substitute the quadratic equation into the equation $y=x^2+4x$ and simplify.

Tips and Tricks

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• When solving equations involving square roots, it's essential to isolate the square root term on one side of the equation.
• Squaring both sides of the equation can help eliminate the square root term and simplify the equation.
• Be careful when subtracting or adding terms to both sides of the equation, as this can affect the final form of the equation.
• If you have a negative value for $x$, you will need to take the absolute value of $x$ before substituting it into the equation.

Frequently Asked Questions

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Q: What is the purpose of squaring both sides of the equation?

A: Squaring both sides of the equation allows us to eliminate the square root term and simplify the equation.

Q: How do I know when to square both sides of the equation?

A: You should square both sides of the equation when you have a square root term on one side of the equation.

Q: What is the final form of the equation?

A: The final form of the equation is $y=x^2+4x$.

Q: How do I solve the equation for a specific value of $x$?

A: To solve the equation for a specific value of $x$, substitute the value of $x$ into the equation $y=x^2+4x$ and simplify.

References

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• [1] Algebraic Manipulations. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/manipulations.html
• [2] Square Root. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/square-root.html
• [3] Quadratic Equations. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/quadratic-equations.html

Note: The references provided are for general information purposes only and are not specific to the equation $\sqrt{y+4}=x+2$.