What Is The Solution To The Equation 6 X − 3 = 60 6 \sqrt{x-3} = 60 6 X − 3 ​ = 60 ?A. X = 17 X = 17 X = 17 B. X = 23 X = 23 X = 23 C. X = 97 X = 97 X = 97 D. X = 103 X = 103 X = 103

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Introduction

Mathematics is a vast and fascinating subject that encompasses various branches, including algebra, geometry, and calculus. One of the fundamental concepts in mathematics is solving equations, which involves finding the value of a variable that satisfies a given equation. In this article, we will focus on solving a specific equation involving a square root, and we will explore the step-by-step process to find the solution.

Understanding the Equation

The given equation is 6x−3=606 \sqrt{x-3} = 60. This equation involves a square root, which is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value. In this case, the square root of x−3x-3 is being multiplied by 6, and the result is equal to 60.

Step 1: Isolate the Square Root

To solve the equation, we need to isolate the square root term. We can do this by dividing both sides of the equation by 6. This will give us:

x−3=606\sqrt{x-3} = \frac{60}{6}

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by dividing 60 by 6, which gives us:

x−3=10\sqrt{x-3} = 10

Step 3: Square Both Sides

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

(x−3)2=102(\sqrt{x-3})^2 = 10^2

Step 4: Simplify the Equation

Squaring both sides of the equation gives us:

x−3=100x-3 = 100

Step 5: Add 3 to Both Sides

To isolate the variable x, we need to add 3 to both sides of the equation. This will give us:

x=100+3x = 100 + 3

Step 6: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by adding 100 and 3, which gives us:

x=103x = 103

Conclusion

In this article, we have solved the equation 6x−3=606 \sqrt{x-3} = 60 using a step-by-step process. We have isolated the square root term, simplified the right-hand side, squared both sides, and finally isolated the variable x. The solution to the equation is x = 103.

Discussion

The solution to the equation 6x−3=606 \sqrt{x-3} = 60 is x = 103. This means that when we substitute x = 103 into the original equation, we get:

6103−3=6100=6⋅10=606 \sqrt{103-3} = 6 \sqrt{100} = 6 \cdot 10 = 60

This confirms that the solution x = 103 is correct.

Final Answer

The final answer to the equation 6x−3=606 \sqrt{x-3} = 60 is x = 103.

Comparison with Other Options

Let's compare the solution x = 103 with the other options:

  • A. x = 17: This is not the correct solution, as we have already shown that x = 103 is the correct solution.
  • B. x = 23: This is not the correct solution, as we have already shown that x = 103 is the correct solution.
  • C. x = 97: This is not the correct solution, as we have already shown that x = 103 is the correct solution.

Conclusion

In conclusion, the solution to the equation 6x−3=606 \sqrt{x-3} = 60 is x = 103. This is the correct solution, and it can be verified by substituting x = 103 into the original equation.

Introduction

Solving equations with square roots can be a challenging task, but with the right approach and techniques, it can be made easier. In this article, we will answer some frequently asked questions (FAQs) 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 term. This can be done by moving all other terms to the other side of the equation.

Q: How do I simplify the right-hand side of the equation after isolating the square root term?

A: After isolating the square root term, you can simplify the right-hand side of the equation by performing any necessary operations, such as adding or subtracting numbers.

Q: What is the next step after simplifying the right-hand side of the equation?

A: The next step is to square both sides of the equation. This will eliminate the square root and allow you to solve for the variable.

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. This is because the square root operation is the inverse of the squaring operation, so by squaring both sides, you can "undo" the square root operation.

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 term first
  • Not simplifying the right-hand side of the equation
  • Not squaring both sides of the equation
  • Not checking the solution to make sure it satisfies the original equation

Q: How do I check the solution to make sure it satisfies the original equation?

A: To check the solution, you can substitute the value of the variable back into the original equation and see if it is true. If the equation is true, then the solution is correct.

Q: What if I get a negative value when I square both sides of the equation?

A: If you get a negative value when you square both sides of the equation, it means that the original equation has no solution. This is because the square of a number cannot be negative.

Q: Can I use a calculator to solve equations with square roots?

A: Yes, you can use a calculator to solve equations with square roots. However, it's always a good idea to check the solution by hand to make sure it's 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 used to calculate distances, velocities, and accelerations.
  • Engineering: Solving equations with square roots is used to design and optimize systems, such as bridges and buildings.
  • Finance: Solving equations with square roots is used to calculate interest rates and investment returns.

Conclusion

Solving equations with square roots can be a challenging task, but with the right approach and techniques, it can be made easier. By following the steps outlined in this article and avoiding common mistakes, you can solve equations with square roots with confidence.