Solve For \[$ X \$\].$\[ \sqrt{x+8} = 2 + X \\]

Introduction

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving equations that involve square roots, specifically the equation $\sqrt{x+8} = 2 + x$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is $\sqrt{x+8} = 2 + x$. To solve this equation, we need to isolate the variable $x$. The first step is to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of $x+8$ is equal to $2 + x$.

Step 1: Square Both Sides

To eliminate the square root, we can square both sides of the equation. This will give us $(\sqrt{x+8})^2 = (2 + x)^2$. Simplifying both sides, we get $x + 8 = 4 + 4x + x^2$.

Step 2: Simplify the Equation

Now, we need to simplify the equation by combining like terms. We can start by subtracting $x$ from both sides, which gives us $8 = 4 + 4x + x^2 - x$. Simplifying further, we get $8 = 4 + 3x + x^2$.

Step 3: Rearrange the Equation

Next, we need to rearrange the equation to get all the terms on one side. We can do this by subtracting $4$ from both sides, which gives us $4 = 3x + x^2$. Now, we need to move all the terms to the left-hand side by subtracting $3x$ from both sides. This gives us $4 - 3x = x^2$.

Step 4: Rearrange the Equation (Again)

Now, we need to rearrange the equation again to get it in a more familiar form. We can do this by subtracting $4$ from both sides, which gives us $-3x = x^2 - 4$. Now, we need to move all the terms to the left-hand side by subtracting $x^2$ from both sides. This gives us $-3x - x^2 = -4$.

Step 5: Factor the Equation

Now, we need to factor the equation to make it easier to solve. We can start by factoring out $-1$ from both terms, which gives us $-(3x + x^2) = -4$. Now, we can factor the quadratic expression inside the parentheses, which gives us $-x(3 + x) = -4$.

Step 6: Solve for x

Now, we need to solve for $x$. We can start by dividing both sides of the equation by $-1$, which gives us $x(3 + x) = 4$. Now, we can expand the left-hand side of the equation, which gives us $3x + x^2 = 4$. Now, we can rearrange the equation to get it in a more familiar form, which gives us $x^2 + 3x - 4 = 0$.

Step 7: Solve the Quadratic Equation

Now, we need to solve the quadratic equation $x^2 + 3x - 4 = 0$. We can use the quadratic formula to solve this equation, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = 3$, and $c = -4$. Plugging these values into the quadratic formula, we get $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)}$. Simplifying further, we get $x = \frac{-3 \pm \sqrt{25}}{2}$. Now, we can simplify the square root, which gives us $x = \frac{-3 \pm 5}{2}$.

Step 8: Find the Solutions

Now, we need to find the solutions to the equation. We can do this by plugging in the values of $x$ into the original equation. We get two possible solutions: $x = \frac{-3 + 5}{2}$ and $x = \frac{-3 - 5}{2}$. Simplifying further, we get $x = 1$ and $x = -4$.

Conclusion

Introduction

In our previous article, we solved the equation $\sqrt{x+8} = 2 + x$ using a step-by-step approach. However, we know that solving equations with square roots can be a challenging task, and many students struggle with it. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving equations with square roots.

Q: What is the first step in solving an equation with a square root?

A: The first step in solving an equation with a square root is to square both sides of the equation. This will eliminate the square root and allow you to simplify the equation.

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

A: We square both sides of the equation to eliminate the square root. When we square both sides, we are essentially getting rid of the square root sign and simplifying the equation.

Q: What is the next step after squaring both sides of the equation?

A: After squaring both sides of the equation, we need to simplify the resulting equation. This involves combining like terms and rearranging the equation to get it in a more familiar form.

Q: How do we simplify the equation after squaring both sides?

A: We simplify the equation by combining like terms and rearranging the equation to get it in a more familiar form. This may involve moving all the terms to one side of the equation or factoring the equation.

Q: What is the quadratic formula, and how do we use it to solve equations with square roots?

A: The quadratic formula is a formula that allows us to solve quadratic equations of the form $ax^2 + bx + c = 0$. To use the quadratic formula, we need to plug in the values of $a$, $b$, and $c$ into the formula and simplify.

Q: How do we use the quadratic formula to solve the equation $\sqrt{x+8} = 2 + x$?

A: To use the quadratic formula to solve the equation $\sqrt{x+8} = 2 + x$, we need to square both sides of the equation and simplify the resulting equation. We then plug in the values of $a$, $b$, and $c$ into the quadratic formula and simplify.

Q: What are the two possible solutions to the equation $\sqrt{x+8} = 2 + x$?

A: The two possible solutions to the equation $\sqrt{x+8} = 2 + x$ are $x = 1$ and $x = -4$.

Q: How do we check our solutions to make sure they are correct?

A: To check our solutions, we need to plug them back into the original equation and make sure they satisfy the equation. If they do, then we know that they are correct.

Q: What are some common mistakes to avoid when solving equations with square roots?

A: Some common mistakes to avoid when solving equations with square roots include:

• Squaring both sides of the equation without checking if the equation is valid
• Simplifying the equation incorrectly
• Using the quadratic formula incorrectly
• Not checking the solutions to make sure they are correct

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in solving equations with square roots. We covered topics such as squaring both sides of the equation, simplifying the equation, using the quadratic formula, and checking solutions. By following these steps and avoiding common mistakes, you can become proficient in solving equations with square roots.

Additional Resources

If you need additional help or resources, here are some suggestions:

• Khan Academy: Solving Equations with Square Roots
• Mathway: Solving Equations with Square Roots
• Wolfram Alpha: Solving Equations with Square Roots

Remember, practice makes perfect! The more you practice solving equations with square roots, the more comfortable you will become with the concepts and techniques involved.