Violet Was Carrying Around The Steps To Solve $\sqrt{3x+1}=4$. On Her Way Through The Math Factory, Violet Dropped Them And They Mixed With Other Steps. Put The Correct Steps In Order To Solve

Feb 27, 2025 by ADMIN 193 views

**Solving Equations with Square Roots: A Step-by-Step Guide**

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 guide you through the steps to solve the equation $\sqrt{3x+1}=4$ . This equation involves a square root, and we will use algebraic manipulation to isolate the variable $x$ .

Step 1: Square Both Sides of the Equation

The first step in solving the equation is to square both sides of the equation. This will eliminate the square root sign and allow us to work with a simpler equation.

$\sqrt{3x+1}=4$

Squaring both sides of the equation gives us:

$3x+1=16$

Step 2: Subtract 1 from Both Sides of the Equation

Now that we have squared both sides of the equation, we can simplify it by subtracting 1 from both sides.

$3x+1=16$

Subtracting 1 from both sides gives us:

$3x=15$

Step 3: Divide Both Sides of the Equation by 3

Now that we have isolated the term with the variable $x$ , we can divide both sides of the equation by 3 to solve for $x$ .

$3x=15$

Dividing both sides of the equation by 3 gives us:

$x=5$

Step 4: Check the Solution

Now that we have solved for $x$ , we can check our solution by plugging it back into the original equation.

$\sqrt{3x+1}=4$

Substituting $x=5$ into the equation gives us:

$\sqrt{3(5)+1}=\sqrt{16}=4$

Since the left-hand side of the equation is equal to the right-hand side, we can conclude that our solution is correct.

Conclusion

Solving equations with square roots requires careful algebraic manipulation. By following the steps outlined in this article, we can solve equations of the form $\sqrt{ax+b}=c$ . Remember to square both sides of the equation, simplify, and check your solution to ensure that it is correct.

Common Mistakes to Avoid

When solving equations with square roots, there are several common mistakes to avoid. These include:

Not squaring both sides of the equation: Failing to square both sides of the equation can lead to incorrect solutions.
Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
Not checking the solution: Failing to check the solution can lead to incorrect answers.

Real-World Applications

Solving equations with square roots has many real-world applications. For example, in physics, the equation $\sqrt{v^2-u^2}=t$ is used to calculate the time it takes for an object to travel a certain distance. In engineering, the equation $\sqrt{a^2+b^2}=c$ is used to calculate the length of a diagonal in a right triangle.

Practice Problems

Solving equations with square roots requires practice. Here are a few practice problems to try:

$\sqrt{x+2}=3$
$\sqrt{2x-1}=5$
$\sqrt{x-3}=2$

Conclusion

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 sign and allow us to work with a simpler equation.

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 sign. If we don't square both sides, we will be left with an equation that contains a square root, which can be difficult to solve.

Q: How do I know if I have squared both sides of the equation correctly?

A: To check if you have squared both sides of the equation correctly, you can plug the squared expression back into the original equation and see if it is true. If it is true, then you have squared both sides correctly.

Q: What if I have a negative number under the square root sign?

A: If you have a negative number under the square root sign, you will need to use the imaginary unit, i, to represent the square root of the negative number. For example, if you have the equation $\sqrt{-x}=3$ , you can rewrite it as $i\sqrt{x}=-3$ .

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, keep in mind that calculators may not always give you the exact solution, and you may need to round your answer.

Q: How do I check my solution to an equation with a square root?

A: To check your solution to an equation with a square root, you can plug the solution back into the original equation and see if it is true. If it is true, then your solution is correct.

Q: What if I get a negative solution to an equation with a square root?

A: If you get a negative solution to an equation with a square root, you will need to check if the solution is extraneous. This means that you will need to plug the solution back into the original equation and see if it is true. If it is not true, then the solution is extraneous and you will need to try again.

Q: Can I use the quadratic formula to solve equations with square roots?

A: Yes, you can use the quadratic formula to solve equations with square roots. However, keep in mind that the quadratic formula is only applicable to quadratic equations, and you will need to use a different method to solve equations with square roots.

Q: How do I know if an equation with a square root has a real solution?

A: To determine if an equation with a square root has a real solution, you can check if the expression under the square root sign is non-negative. If it is non-negative, then the equation has a real solution. If it is negative, then the equation does not have a real solution.

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

A: Yes, you can use a graphing calculator to solve equations with square roots. However, keep in mind that graphing calculators may not always give you the exact solution, and you may need to round your answer.

Conclusion

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article, you can solve equations of the form $\sqrt{ax+b}=c$ . Remember to square both sides of the equation, simplify, and check your solution to ensure that it is correct. With practice, you will become proficient in solving equations with square roots.