Solve For \[$ X \$\].$\[ 3 \sqrt{2x - 1} = 3x \\]

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Introduction

Radical equations are a type of algebraic equation that involves a variable within a radical expression. In this article, we will focus on solving radical equations of the form a=b\sqrt{a} = b, where aa and bb are expressions involving variables. We will use the given equation 32x−1=3x3\sqrt{2x - 1} = 3x as an example to demonstrate the steps involved in solving radical equations.

Understanding Radical Equations

Radical equations are equations that contain a variable within a radical expression. The radical expression can be a square root, cube root, or any other root. Radical equations can be solved using various techniques, including isolating the variable, squaring both sides, and simplifying the resulting expression.

Step 1: Isolate the Radical Expression

The first step in solving a radical equation is to isolate the radical expression on one side of the equation. In the given equation 32x−1=3x3\sqrt{2x - 1} = 3x, we can start by dividing both sides of the equation by 3 to isolate the radical expression.

32x−13=3x3\frac{3\sqrt{2x - 1}}{3} = \frac{3x}{3}

This simplifies to:

2x−1=x\sqrt{2x - 1} = x

Step 2: Square Both Sides

The next step is to square both sides of the equation to eliminate the radical expression. When we square both sides of the equation, we must remember to square both the left-hand side and the right-hand side of the equation.

(2x−1)2=x2(\sqrt{2x - 1})^2 = x^2

This simplifies to:

2x−1=x22x - 1 = x^2

Step 3: Simplify the Resulting Expression

Now that we have squared both sides of the equation, we can simplify the resulting expression. We can start by rearranging the terms to get a quadratic equation in standard form.

x2−2x+1=0x^2 - 2x + 1 = 0

This is a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0, where a=1a = 1, b=−2b = -2, and c=1c = 1.

Step 4: Solve the Quadratic Equation

We can solve the quadratic equation using various techniques, including factoring, completing the square, or using the quadratic formula. In this case, we can use the quadratic formula to find the solutions.

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting the values of aa, bb, and cc, we get:

x=−(−2)±(−2)2−4(1)(1)2(1)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(1)}}{2(1)}

This simplifies to:

x=2±4−42x = \frac{2 \pm \sqrt{4 - 4}}{2}

x=2±02x = \frac{2 \pm \sqrt{0}}{2}

x=22x = \frac{2}{2}

x=1x = 1

Conclusion

Introduction

Radical equations can be a challenging topic for many students. In our previous article, we provided a step-by-step guide on how to solve radical equations. In this article, we will answer some of the most frequently asked questions about solving radical equations.

Q: What is a radical equation?

A: A radical equation is an equation that contains a variable within a radical expression. The radical expression can be a square root, cube root, or any other root.

Q: How do I know if an equation is a radical equation?

A: To determine if an equation is a radical equation, look for the presence of a radical symbol (such as √) or a power of a fraction (such as 1/2). If you see either of these, it's likely a radical equation.

Q: What are the steps to solve a radical equation?

A: The steps to solve a radical equation are:

  1. Isolate the radical expression on one side of the equation.
  2. Square both sides of the equation to eliminate the radical expression.
  3. Simplify the resulting expression.
  4. Solve the resulting 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 radical expression. When you square both sides, you are essentially getting rid of the radical sign.

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

A: If you have a negative number under the radical sign, you will need to use the absolute value of the number. For example, if you have √(-x), you would rewrite it as √|x|.

Q: Can I use a calculator to solve radical equations?

A: While a calculator can be helpful in solving radical equations, it's not always the best approach. Calculators can sometimes give you incorrect answers or round numbers, which can lead to errors. It's usually best to solve radical equations by hand.

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

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

  • Not isolating the radical expression on one side of the equation.
  • Not squaring both sides of the equation.
  • Not simplifying the resulting expression.
  • Not checking for extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, plug the solution back into the original equation and see if it's true. If it's not true, then the solution is extraneous.

Q: Can I use the quadratic formula to solve radical equations?

A: Yes, you can use the quadratic formula to solve radical equations. However, you will need to first simplify the equation and then apply the quadratic formula.

Conclusion

Solving radical equations can be a challenging topic, but with practice and patience, you can master it. Remember to isolate the radical expression, square both sides of the equation, simplify the resulting expression, and solve the resulting equation. Don't be afraid to ask for help if you get stuck, and always check for extraneous solutions.