What Is The Solution To The Equation Below?$\sqrt{x+3}=x-3$A. $x=4$B. $x=6$C. $x=3$D. $x=1$

Mar 1, 2025 by ADMIN 92 views

**Solving the Equation: A Step-by-Step Guide**

Introduction

Equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific equation: $\sqrt{x+3}=x-3$ . This equation involves a square root, which can be challenging to solve. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

Before we dive into the solution, let's understand the equation. The equation is $\sqrt{x+3}=x-3$ , where $x$ is the variable we need to solve for. The square root of $x+3$ is equal to $x-3$ . Our goal is to find the value of $x$ that satisfies this equation.

Step 1: Isolate the Square Root

To solve the equation, we need to isolate the square root. We can do this by squaring both sides of the equation. This will eliminate the square root and allow us to solve for $x$ .

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

Step 2: Expand the Squared Terms

Now that we have squared both sides of the equation, we need to expand the squared terms.

$x+3 = x^2 - 6x + 9$

Step 3: Simplify the Equation

We can simplify the equation by moving all the terms to one side.

$x^2 - 6x + 9 - x - 3 = 0$

$x^2 - 7x + 6 = 0$

Step 4: Factor the Quadratic Equation

The equation is now a quadratic equation, which can be factored.

$(x - 1)(x - 6) = 0$

Step 5: Solve for x

We can now solve for $x$ by setting each factor equal to zero.

$x - 1 = 0$ or $x - 6 = 0$

$x = 1$ or $x = 6$

Conclusion

We have solved the equation $\sqrt{x+3}=x-3$ and found two possible values for $x$ : $x = 1$ and $x = 6$ . To determine which value is correct, we need to substitute each value back into the original equation and check if it is true.

Checking the Solutions

Let's substitute $x = 1$ into the original equation.

$\sqrt{1+3} = 1 - 3$

$\sqrt{4} = -2$

This is not true, so $x = 1$ is not a valid solution.

Now, let's substitute $x = 6$ into the original equation.

$\sqrt{6+3} = 6 - 3$

$\sqrt{9} = 3$

This is true, so $x = 6$ is a valid solution.

Final Answer

The final answer is $\boxed{6}$ .

Additional Tips and Tricks

When solving equations with square roots, it's essential to isolate the square root first.
Squaring both sides of the equation can help eliminate the square root.
Factoring quadratic equations can make it easier to solve for $x$ .
Always check the solutions by substituting them back into the original equation.

Common Mistakes to Avoid

Not isolating the square root before squaring both sides of the equation.
Not expanding the squared terms correctly.
Not simplifying the equation properly.
Not factoring the quadratic equation correctly.

Introduction

In our previous article, we solved the equation $\sqrt{x+3}=x-3$ and found the value of $x$ to be $6$ . However, we know that there are often multiple ways to solve a problem, and different people may have different approaches. In this article, we will answer some common questions about 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 isolate the square root. This means getting the square root term by itself on one side of the equation.

Q: How do I isolate the square root?

A: To isolate the square root, you can use inverse operations. For example, if the square root is being added to another term, you can subtract that term from both sides of the equation. If the square root is being multiplied by another term, you can divide both sides of the equation by that term.

Q: What is the next step after isolating the square root?

A: After isolating the square root, the next step is to square both sides of the equation. This will eliminate the square root and allow you to solve for $x$ .

Q: Why do I need to square both sides of the equation?

A: Squaring both sides of the equation is necessary to eliminate the square root. When you square both sides, you are essentially getting rid of the square root sign and simplifying the equation.

Q: How do I expand the squared terms?

A: To expand the squared terms, you need to multiply each term inside the parentheses by itself. For example, if you have $(x + 3)^2$ , you would multiply $x$ by $x$ and $3$ by $3$ , and then add the results.

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

A: The final step in solving an equation with a square root is to solve for $x$ . This may involve factoring the quadratic equation, using the quadratic formula, or other methods.

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:

Not isolating the square root before squaring both sides of the equation.
Not expanding the squared terms correctly.
Not simplifying the equation properly.
Not factoring the quadratic equation correctly.

Q: How do I check my solutions?

A: To check your solutions, you need to substitute each value back into the original equation and check if it is true. This will help you ensure that your solution is correct.

Q: What are some real-world applications of solving equations with square roots?

A: Solving equations with square roots has many real-world applications, including:

Physics: Solving equations with square roots is essential in physics, where you may need to calculate distances, velocities, and accelerations.
Engineering: Solving equations with square roots is also important in engineering, where you may need to design and build structures, machines, and systems.
Computer Science: Solving equations with square roots is used in computer science, where you may need to optimize algorithms and solve complex problems.

Conclusion

Solving equations with square roots can be challenging, but with practice and patience, you can master this skill. Remember to isolate the square root, square both sides of the equation, expand the squared terms, and solve for $x$ . By following these steps and avoiding common mistakes, you can solve equations with square roots like a pro.