Solve The Equation:${ (\sqrt{3x+3})^2 = (x+1)^2 }$

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Introduction

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. In this article, we will focus on solving the equation $(\sqrt{3x+3})^2 = (x+1)^2$. This equation involves square roots and quadratic expressions, making it a bit more challenging than a simple linear equation. However, with the right approach and techniques, we can solve this equation and find the values of x that satisfy it.

Understanding the Equation

The given equation is $(\sqrt{3x+3})^2 = (x+1)^2$. The first step in solving this equation is to understand what it means. The expression $(\sqrt{3x+3})^2$ represents the square of the square root of $3x+3$. Similarly, the expression $(x+1)^2$ represents the square of $x+1$. Our goal is to find the values of x that make these two expressions equal.

Simplifying the Equation

To simplify the equation, we can start by expanding the squared expressions. Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we can expand the left-hand side of the equation as follows:

$(\sqrt{3x+3})^2 = 3x + 3$

Similarly, we can expand the right-hand side of the equation as follows:

$(x+1)^2 = x^2 + 2x + 1$

Now that we have expanded both sides of the equation, we can set them equal to each other:

$3x + 3 = x^2 + 2x + 1$

Rearranging the Equation

The next step is to rearrange the equation to get all the terms on one side. We can do this by subtracting $3x$ from both sides of the equation:

$3 = x^2 + 2x - 3x + 1$

Simplifying further, we get:

$3 = x^2 - x + 1$

Bringing All Terms to One Side

Now that we have all the terms on one side of the equation, we can simplify it further by combining like terms. We can do this by subtracting 1 from both sides of the equation:

$2 = x^2 - x$

Rearranging the Equation Again

The next step is to rearrange the equation to get it in a more familiar form. We can do this by subtracting $x^2$ from both sides of the equation:

$-x^2 + 2 = -x$

Bringing All Terms to One Side Again

Now that we have all the terms on one side of the equation, we can simplify it further by combining like terms. We can do this by adding $x^2$ to both sides of the equation:

$-x^2 + x^2 + 2 = -x + x^2$

Simplifying further, we get:

$2 = x^2 - x$

Factoring the Equation

The next step is to factor the equation. We can do this by factoring out a common term from both terms on the right-hand side of the equation:

$2 = (x - 1)(x + 2)$

Finding the Values of x

Now that we have factored the equation, we can find the values of x that satisfy it. We can do this by setting each factor equal to zero and solving for x:

$x - 1 = 0 \Rightarrow x = 1$

$x + 2 = 0 \Rightarrow x = -2$

Conclusion

In this article, we have solved the equation $(\sqrt{3x+3})^2 = (x+1)^2$. We started by simplifying the equation and then rearranging it to get all the terms on one side. We then factored the equation and found the values of x that satisfy it. The values of x that satisfy the equation are x = 1 and x = -2.

Final Answer

The final answer is $\boxed{1, -2}$.

Discussion

The equation $(\sqrt{3x+3})^2 = (x+1)^2$ is a quadratic equation that involves square roots and quadratic expressions. Solving this equation requires a combination of algebraic techniques, including simplifying and rearranging the equation, factoring, and finding the values of x that satisfy it. The values of x that satisfy the equation are x = 1 and x = -2.

Related Topics

• Solving quadratic equations
• Simplifying and rearranging equations
• Factoring equations
• Finding the values of x that satisfy an equation

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information purposes only and are not specific to the equation $(\sqrt{3x+3})^2 = (x+1)^2$.

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Introduction

In our previous article, we solved the equation $(\sqrt{3x+3})^2 = (x+1)^2$ and found the values of x that satisfy it. In this article, we will answer some frequently asked questions related to this equation.

Q: What is the main concept behind solving this equation?

A: The main concept behind solving this equation is to simplify and rearrange the equation to get all the terms on one side, and then factor the equation to find the values of x that satisfy it.

Q: What are the steps involved in solving this equation?

A: The steps involved in solving this equation are:

1. Simplifying the equation by expanding the squared expressions
2. Rearranging the equation to get all the terms on one side
3. Factoring the equation to find the values of x that satisfy it

Q: What are the values of x that satisfy this equation?

A: The values of x that satisfy this equation are x = 1 and x = -2.

Q: How do I know which values of x satisfy the equation?

A: To determine which values of x satisfy the equation, you need to set each factor equal to zero and solve for x. In this case, we set x - 1 = 0 and x + 2 = 0, and solved for x.

Q: What if I get stuck while solving the equation?

A: If you get stuck while solving the equation, try simplifying and rearranging the equation again, or factoring it in a different way. You can also try using different algebraic techniques, such as using the quadratic formula.

Q: Can I use this equation in real-life situations?

A: Yes, this equation can be used in real-life situations, such as in physics, engineering, and computer science. For example, you can use this equation to model the motion of an object, or to solve optimization problems.

Q: How do I apply this equation to a real-life problem?

A: To apply this equation to a real-life problem, you need to identify the variables and constants in the equation, and then substitute the values of the variables and constants into the equation. You can then solve for the unknown variable.

Q: What are some common mistakes to avoid while solving this equation?

A: Some common mistakes to avoid while solving this equation include:

• Not simplifying and rearranging the equation enough
• Not factoring the equation correctly
• Not checking the solutions for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solutions back into the original equation and check if they are true. If they are not true, then they are extraneous solutions.

Q: What are some tips for solving this equation?

A: Some tips for solving this equation include:

• Simplifying and rearranging the equation as much as possible
• Factoring the equation in a way that makes it easy to solve
• Checking the solutions for extraneous solutions

Q: Can I use this equation to solve other types of equations?

A: Yes, this equation can be used to solve other types of equations, such as quadratic equations and polynomial equations.

Q: How do I apply this equation to other types of equations?

A: To apply this equation to other types of equations, you need to identify the variables and constants in the equation, and then substitute the values of the variables and constants into the equation. You can then solve for the unknown variable.

Conclusion

In this article, we have answered some frequently asked questions related to the equation $(\sqrt{3x+3})^2 = (x+1)^2$. We have also provided some tips and advice for solving this equation, and for applying it to real-life situations.

Final Answer

The final answer is $\boxed{1, -2}$.

Discussion

The equation $(\sqrt{3x+3})^2 = (x+1)^2$ is a quadratic equation that involves square roots and quadratic expressions. Solving this equation requires a combination of algebraic techniques, including simplifying and rearranging the equation, factoring, and finding the values of x that satisfy it. The values of x that satisfy the equation are x = 1 and x = -2.

Related Topics

• Solving quadratic equations
• Simplifying and rearranging equations
• Factoring equations
• Finding the values of x that satisfy an equation

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information purposes only and are not specific to the equation $(\sqrt{3x+3})^2 = (x+1)^2$.