Find All Real Number Solutions For The Equation:$\[ X + 7x^{\frac{1}{2}} - 18 = 0 \\]If There Is More Than One Solution, Separate Them With A Comma:$\[ X = \\]

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Introduction

In this article, we will delve into the world of algebra and solve the equation $x + 7x^{\frac{1}{2}} - 18 = 0$. This equation involves a square root term, which makes it a bit more challenging to solve than a standard quadratic equation. We will use various techniques to isolate the variable $x$ and find all real number solutions.

Understanding the Equation

The given equation is $x + 7x^{\frac{1}{2}} - 18 = 0$. To begin solving this equation, we need to understand the properties of square roots. The square root of a number $x$ is a value that, when multiplied by itself, gives $x$. In other words, $x^{\frac{1}{2}}$ is the value that, when squared, gives $x$.

Isolating the Square Root Term

To solve the equation, we need to isolate the square root term. We can do this by moving the constant term to the right-hand side of the equation. This gives us:

$x + 7x^{\frac{1}{2}} = 18$

Squaring Both Sides

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

$(x + 7x^{\frac{1}{2}})^2 = 18^2$

Expanding the Left-Hand Side

Expanding the left-hand side of the equation, we get:

$x^2 + 14x^{\frac{1}{2}} + 49 = 324$

Rearranging the Terms

Rearranging the terms, we get:

$x^2 + 14x^{\frac{1}{2}} - 275 = 0$

Substituting $y = x^{\frac{1}{2}}$

To simplify the equation, we can substitute $y = x^{\frac{1}{2}}$. This gives us:

$x^2 = y^2$

Substituting this into the equation, we get:

$y^2 + 14y - 275 = 0$

Factoring the Quadratic Equation

The quadratic equation $y^2 + 14y - 275 = 0$ can be factored as:

$(y + 25)(y - 11) = 0$

Solving for $y$

Solving for $y$, we get:

$y + 25 = 0$ or $y - 11 = 0$

Solving for $x$

Substituting $y = x^{\frac{1}{2}}$ back into the equations, we get:

$x^{\frac{1}{2}} + 25 = 0$ or $x^{\frac{1}{2}} - 11 = 0$

Solving for $x$, we get:

$x^{\frac{1}{2}} = -25$ or $x^{\frac{1}{2}} = 11$

Squaring Both Sides

Squaring both sides of the equations, we get:

$x = (-25)^2$ or $x = (11)^2$

Simplifying the Equations

Simplifying the equations, we get:

$x = 625$ or $x = 121$

Conclusion

In this article, we solved the equation $x + 7x^{\frac{1}{2}} - 18 = 0$ using various techniques. We isolated the square root term, squared both sides, and substituted $y = x^{\frac{1}{2}}$ to simplify the equation. We then factored the quadratic equation, solved for $y$, and substituted $y = x^{\frac{1}{2}}$ back into the equations to solve for $x$. The final solutions are $x = 625$ and $x = 121$.

Final Answer

The final answer is: $\boxed{625, 121}$

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Introduction

In our previous article, we solved the equation $x + 7x^{\frac{1}{2}} - 18 = 0$ using various techniques. In this article, we will answer some frequently asked questions related to the solution of this equation.

Q: What is the first step in solving the equation $x + 7x^{\frac{1}{2}} - 18 = 0$?

A: The first step in solving the equation $x + 7x^{\frac{1}{2}} - 18 = 0$ is to isolate the square root term. We can do this by moving the constant term to the right-hand side of the equation.

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. Squaring both sides of the equation allows us to get rid of the square root term and solve for $x$.

Q: What is the significance of substituting $y = x^{\frac{1}{2}}$?

A: Substituting $y = x^{\frac{1}{2}}$ simplifies the equation and allows us to solve for $y$. Once we have solved for $y$, we can substitute $y = x^{\frac{1}{2}}$ back into the equation to solve for $x$.

Q: How do we solve the quadratic equation $y^2 + 14y - 275 = 0$?

A: We can solve the quadratic equation $y^2 + 14y - 275 = 0$ by factoring it. The equation can be factored as $(y + 25)(y - 11) = 0$. We can then solve for $y$ by setting each factor equal to zero.

Q: What are the final solutions to the equation $x + 7x^{\frac{1}{2}} - 18 = 0$?

A: The final solutions to the equation $x + 7x^{\frac{1}{2}} - 18 = 0$ are $x = 625$ and $x = 121$.

Q: Why do we need to check our solutions?

A: We need to check our solutions to make sure that they are valid. In this case, we need to check that the solutions $x = 625$ and $x = 121$ satisfy the original equation.

Q: How do we check that the solutions are valid?

A: We can check that the solutions are valid by substituting them back into the original equation. If the solutions satisfy the original equation, then they are valid.

Conclusion

In this article, we answered some frequently asked questions related to the solution of the equation $x + 7x^{\frac{1}{2}} - 18 = 0$. We covered topics such as isolating the square root term, squaring both sides of the equation, substituting $y = x^{\frac{1}{2}}$, solving the quadratic equation, and checking the solutions.

Final Answer

The final answer is: $\boxed{625, 121}$

Additional Resources

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

• Solving Equations with Square Roots
• Solving Quadratic Equations
• Substituting Variables

Related Articles

For more information on solving equations, please see the following related articles:

• Solving Linear Equations
• Solving Quadratic Equations
• Solving Systems of Equations