Select The Correct Answer.What Is The Solution To The Equation?$\[ \sqrt{x-3}+1=6 \\]A. 64 B. 46 C. 28 D. 22

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, $\sqrt{x-3}+1=6$, and provide a step-by-step guide to help you arrive at the correct solution.

Understanding the Equation

The given equation is $\sqrt{x-3}+1=6$. To solve this equation, we need to isolate the variable $x$. The equation involves a square root, which can be challenging to work with. However, with a systematic approach, we can simplify the equation and find the value of $x$.

Step 1: Isolate the Square Root

The first step is to isolate the square root term on one side of the equation. We can do this by subtracting 1 from both sides of the equation:

$\sqrt{x-3}+1-1=6-1$

This simplifies to:

$\sqrt{x-3}=5$

Step 2: Square Both Sides

To eliminate the square root, we can square both sides of the equation:

$(\sqrt{x-3})^2=5^2$

This simplifies to:

$x-3=25$

Step 3: Add 3 to Both Sides

To isolate the variable $x$, we need to add 3 to both sides of the equation:

$x-3+3=25+3$

This simplifies to:

$x=28$

Conclusion

In this article, we solved the equation $\sqrt{x-3}+1=6$ using a step-by-step approach. We isolated the square root term, squared both sides, and added 3 to both sides to arrive at the solution. The correct answer is $x=28$.

Why is this Solution Correct?

To verify the solution, we can substitute $x=28$ back into the original equation:

$\sqrt{28-3}+1=6$

This simplifies to:

$\sqrt{25}+1=6$

Which is equal to:

$5+1=6$

This confirms that the solution $x=28$ is correct.

What's Next?

Solving equations is a fundamental skill in mathematics, and there are many more equations to solve. In the next article, we will explore more complex equations and provide additional examples to help you practice and improve your skills.

Common Mistakes to Avoid

When solving equations, it's essential to avoid common mistakes. Here are a few tips to help you:

• Always isolate the variable on one side of the equation.
• Be careful when squaring both sides of the equation, as this can introduce extraneous solutions.
• Verify your solution by substituting it back into the original equation.

Conclusion

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Q: What is the first step in solving an equation?

A: The first step in solving an equation is to isolate the variable on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: How do I deal with square roots in an equation?

A: When dealing with square roots in an equation, you can isolate the square root term by subtracting or adding a value to both sides of the equation. Then, you can square both sides of the equation to eliminate the square root.

Q: What is the difference between an equation and an expression?

A: An equation is a statement that says two expressions are equal, while an expression is a group of numbers, variables, and mathematical operations. For example, $x+3$ is an expression, while $x+3=5$ is an equation.

Q: How do I know if my solution is correct?

A: To verify your solution, you can substitute it back into the original equation. If the equation holds true, then your solution is correct.

Q: What is the order of operations in solving equations?

A: The order of operations in solving equations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I use a calculator to solve equations?

A: Yes, you can use a calculator to solve equations, but it's essential to understand the underlying math concepts and to verify your solution by substituting it back into the original equation.

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

A: Some common mistakes to avoid when solving equations include:

• Not isolating the variable on one side of the equation.
• Squaring both sides of the equation without checking for extraneous solutions.
• Not verifying the solution by substituting it back into the original equation.

Q: How do I solve equations with fractions?

A: To solve equations with fractions, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: Can I solve equations with variables on both sides?

A: Yes, you can solve equations with variables on both sides by adding, subtracting, multiplying, or dividing both sides of the equation to isolate one variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Conclusion

Solving equations is a crucial skill in mathematics, and with practice and patience, you can become proficient in solving even the most complex equations. In this article, we answered some frequently asked questions about solving equations, including how to deal with square roots, how to verify solutions, and how to avoid common mistakes.