Solve The Equation: 2 X 2 + 9 ( X + 1 ) 2 − 27 = 0 2x^2 + 9(\sqrt{x+1})^2 - 27 = 0 2 X 2 + 9 ( X + 1 ​ ) 2 − 27 = 0

Introduction

In this article, we will delve into the world of mathematics and explore a unique equation that involves a square root. The given equation is $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$. Our goal is to solve this equation and find the values of $x$ that satisfy it. We will use various mathematical techniques and strategies to simplify the equation and ultimately find the solutions.

Understanding the Equation

The given equation is a quadratic equation that involves a square root. The square root term is $\sqrt{x+1}$, which is raised to the power of 2. This means that we can simplify the equation by expanding the square of the square root term. We can start by expanding the square of the square root term using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, we have $(\sqrt{x+1})^2 = x+1$.

Simplifying the Equation

Now that we have expanded the square of the square root term, we can simplify the equation. We can start by combining like terms. The equation becomes $2x^2 + 9(x+1) - 27 = 0$. We can simplify this further by distributing the 9 to the terms inside the parentheses. This gives us $2x^2 + 9x + 9 - 27 = 0$. We can then combine the constant terms to get $2x^2 + 9x - 18 = 0$.

Using the Quadratic Formula

The equation $2x^2 + 9x - 18 = 0$ is a quadratic equation in the form $ax^2 + bx + c = 0$. We can use the quadratic formula to solve this equation. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, we have $a = 2$, $b = 9$, and $c = -18$. We can plug these values into the quadratic formula to get $x = \frac{-9 \pm \sqrt{9^2 - 4(2)(-18)}}{2(2)}$.

Solving for x

Now that we have the quadratic formula, we can solve for $x$. We can start by simplifying the expression inside the square root. We have $9^2 - 4(2)(-18) = 81 + 144 = 225$. We can then simplify the expression inside the square root to get $\sqrt{225} = 15$. We can then simplify the expression for $x$ to get $x = \frac{-9 \pm 15}{4}$.

Finding the Solutions

Now that we have the expression for $x$, we can find the solutions. We have $x = \frac{-9 \pm 15}{4}$. We can simplify this further by combining the terms inside the parentheses. We have $x = \frac{-9 + 15}{4}$ or $x = \frac{-9 - 15}{4}$. We can then simplify these expressions to get $x = \frac{6}{4}$ or $x = \frac{-24}{4}$. We can then simplify these expressions further to get $x = \frac{3}{2}$ or $x = -6$.

Checking the Solutions

Now that we have the solutions, we can check them to make sure they satisfy the original equation. We can start by plugging $x = \frac{3}{2}$ into the original equation. We have $2(\frac{3}{2})^2 + 9(\sqrt{\frac{3}{2}+1})^2 - 27 = 0$. We can simplify this expression to get $\frac{9}{2} + 9(\sqrt{\frac{5}{2}})^2 - 27 = 0$. We can then simplify this expression further to get $\frac{9}{2} + 9(\frac{5}{2}) - 27 = 0$. We can then simplify this expression further to get $\frac{9}{2} + \frac{45}{2} - 27 = 0$. We can then simplify this expression further to get $\frac{54}{2} - 27 = 0$. We can then simplify this expression further to get $27 - 27 = 0$. This shows that $x = \frac{3}{2}$ is a solution to the original equation.

Conclusion

In this article, we solved the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$. We used various mathematical techniques and strategies to simplify the equation and ultimately find the solutions. We found that the solutions to the equation are $x = \frac{3}{2}$ and $x = -6$. We then checked these solutions to make sure they satisfy the original equation. We found that $x = \frac{3}{2}$ is a solution to the original equation, but $x = -6$ is not. This shows that the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$ has only one solution, which is $x = \frac{3}{2}$.

Final Answer

The final answer is $\boxed{\frac{3}{2}}$.

Introduction

In our previous article, we solved the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$. We used various mathematical techniques and strategies to simplify the equation and ultimately find the solutions. In this article, we will answer some of the most frequently asked questions about the equation and its solutions.

Q: What is the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$?

A: The equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$ is a quadratic equation that involves a square root. The square root term is $\sqrt{x+1}$, which is raised to the power of 2.

Q: How do I simplify the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$?

A: To simplify the equation, we can start by expanding the square of the square root term using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, we have $(\sqrt{x+1})^2 = x+1$. We can then simplify the equation by combining like terms.

Q: What are the solutions to the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$?

A: The solutions to the equation are $x = \frac{3}{2}$ and $x = -6$. However, we found that $x = -6$ is not a solution to the original equation.

Q: How do I check if a solution satisfies the original equation?

A: To check if a solution satisfies the original equation, we can plug the solution into the original equation and simplify the expression. If the expression equals 0, then the solution satisfies the original equation.

Q: What is the final answer to the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$?

A: The final answer to the equation is $\boxed{\frac{3}{2}}$.

Q: Can I use the quadratic formula to solve the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$?

A: Yes, we can use the quadratic formula to solve the equation. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, we have $a = 2$, $b = 9$, and $c = -18$. We can plug these values into the quadratic formula to get $x = \frac{-9 \pm \sqrt{9^2 - 4(2)(-18)}}{2(2)}$.

Q: What is the significance of the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$?

A: The equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$ is a quadratic equation that involves a square root. It is a good example of how to simplify and solve equations that involve square roots.

Conclusion

In this article, we answered some of the most frequently asked questions about the equation $2x^2 + 9(\sqrt{x+1})^2 - 27 = 0$ and its solutions. We hope that this article has been helpful in understanding the equation and its solutions.

Final Answer

The final answer is $\boxed{\frac{3}{2}}$.