Solve The Radical Equation: X + 2 + 2 X = 18 − X \sqrt{x+2} + \sqrt{2x} = \sqrt{18-x} X + 2 ​ + 2 X ​ = 18 − X ​

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Introduction

Radical equations are a type of algebraic equation that involves a square root or other radical expression. These equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the radical equation x+2+2x=18−x\sqrt{x+2} + \sqrt{2x} = \sqrt{18-x}.

Understanding Radical Equations

Before we dive into solving the equation, let's take a moment to understand what radical equations are and how they work. A radical equation is an equation that contains a square root or other radical expression. The radical expression is typically denoted by the symbol \sqrt{}, and it represents the square root of the expression inside the symbol.

Radical equations can be solved using various methods, including algebraic manipulation, substitution, and elimination. The key to solving radical equations is to isolate the radical expression and then square both sides of the equation to eliminate the radical.

Solving the Radical Equation

Now that we have a basic understanding of radical equations, let's move on to solving the equation x+2+2x=18−x\sqrt{x+2} + \sqrt{2x} = \sqrt{18-x}. To solve this equation, we will use a combination of algebraic manipulation and substitution.

Step 1: Isolate the Radical Expressions

The first step in solving the equation is to isolate the radical expressions on one side of the equation. We can do this by subtracting 2x\sqrt{2x} from both sides of the equation:

x+2=18−x−2x\sqrt{x+2} = \sqrt{18-x} - \sqrt{2x}

Step 2: Square Both Sides of the Equation

Next, we will square both sides of the equation to eliminate the radical. This will give us:

(x+2)2=(18−x−2x)2(\sqrt{x+2})^2 = (\sqrt{18-x} - \sqrt{2x})^2

Expanding the right-hand side of the equation, we get:

x+2=18−x−218−x2x+2xx+2 = 18-x - 2\sqrt{18-x}\sqrt{2x} + 2x

Step 3: Simplify the Equation

Now, let's simplify the equation by combining like terms:

x+2=18−x−218−x2x+2xx+2 = 18-x - 2\sqrt{18-x}\sqrt{2x} + 2x

Subtracting xx from both sides of the equation, we get:

2=18−218−x2x2 = 18 - 2\sqrt{18-x}\sqrt{2x}

Step 4: Isolate the Radical Expression

Next, we will isolate the radical expression by subtracting 18 from both sides of the equation:

−16=−218−x2x-16 = -2\sqrt{18-x}\sqrt{2x}

Dividing both sides of the equation by -2, we get:

8=18−x2x8 = \sqrt{18-x}\sqrt{2x}

Step 5: Square Both Sides of the Equation Again

Finally, we will square both sides of the equation again to eliminate the radical:

(8)2=(18−x2x)2(8)^2 = (\sqrt{18-x}\sqrt{2x})^2

Expanding the right-hand side of the equation, we get:

64=(18−x)(2x)64 = (18-x)(2x)

Step 6: Simplify the Equation

Now, let's simplify the equation by expanding the right-hand side:

64=36x−2x264 = 36x - 2x^2

Rearranging the equation, we get:

2x2−36x+64=02x^2 - 36x + 64 = 0

Step 7: Solve the Quadratic Equation

The final step is to solve the quadratic equation using the quadratic formula:

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

In this case, a=2a = 2, b=−36b = -36, and c=64c = 64. Plugging these values into the quadratic formula, we get:

x=−(−36)±(−36)2−4(2)(64)2(2)x = \frac{-(-36) \pm \sqrt{(-36)^2 - 4(2)(64)}}{2(2)}

Simplifying the equation, we get:

x=36±1296−5124x = \frac{36 \pm \sqrt{1296 - 512}}{4}

x=36±7844x = \frac{36 \pm \sqrt{784}}{4}

x=36±284x = \frac{36 \pm 28}{4}

Therefore, the solutions to the equation are:

x=36+284=16x = \frac{36 + 28}{4} = 16

x=36−284=2x = \frac{36 - 28}{4} = 2

Step 8: Check the Solutions

The final step is to check the solutions by plugging them back into the original equation. If the solutions satisfy the equation, then they are valid solutions.

Plugging x=16x = 16 into the original equation, we get:

16+2+2(16)=18−16\sqrt{16+2} + \sqrt{2(16)} = \sqrt{18-16}

18+32=2\sqrt{18} + \sqrt{32} = \sqrt{2}

This is not true, so x=16x = 16 is not a valid solution.

Plugging x=2x = 2 into the original equation, we get:

2+2+2(2)=18−2\sqrt{2+2} + \sqrt{2(2)} = \sqrt{18-2}

4+4=16\sqrt{4} + \sqrt{4} = \sqrt{16}

2+2=42 + 2 = 4

This is true, so x=2x = 2 is a valid solution.

Conclusion

Introduction

Radical equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will provide a Q&A guide to help you understand and solve radical equations.

Q: What is a radical equation?

A: A radical equation is an equation that contains a square root or other radical expression. The radical expression is typically denoted by the symbol \sqrt{}, and it represents the square root of the expression inside the symbol.

Q: How do I solve a radical equation?

A: To solve a radical equation, you need to isolate the radical expression and then square both sides of the equation to eliminate the radical. This will give you a quadratic equation that you can solve using the quadratic formula.

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 before squaring both sides of the equation
  • Squaring both sides of the equation without isolating the radical expression
  • Not checking the solutions to make sure they satisfy the original equation

Q: How do I check the solutions to a radical equation?

A: To check the solutions to a radical equation, you need to plug the solutions back into the original equation and make sure they satisfy the equation. If the solutions do not satisfy the equation, then they are not valid solutions.

Q: What are some common types of radical equations?

A: Some common types of radical equations include:

  • Equations with a single radical expression
  • Equations with multiple radical expressions
  • Equations with a radical expression and a linear expression
  • Equations with a radical expression and a quadratic expression

Q: How do I solve an equation with a single radical expression?

A: To solve an equation with a single radical expression, you need to isolate the radical expression and then square both sides of the equation to eliminate the radical. This will give you a quadratic equation that you can solve using the quadratic formula.

Q: How do I solve an equation with multiple radical expressions?

A: To solve an equation with multiple radical expressions, you need to isolate one of the radical expressions and then square both sides of the equation to eliminate the radical. This will give you a quadratic equation that you can solve using the quadratic formula. You will then need to repeat the process for the other radical expressions.

Q: How do I solve an equation with a radical expression and a linear expression?

A: To solve an equation with a radical expression and a linear expression, you need to isolate the radical expression and then square both sides of the equation to eliminate the radical. This will give you a quadratic equation that you can solve using the quadratic formula. You will then need to check the solutions to make sure they satisfy the original equation.

Q: How do I solve an equation with a radical expression and a quadratic expression?

A: To solve an equation with a radical expression and a quadratic expression, you need to isolate the radical expression and then square both sides of the equation to eliminate the radical. This will give you a quadratic equation that you can solve using the quadratic formula. You will then need to check the solutions to make sure they satisfy the original equation.

Conclusion

Solving radical equations can be challenging, but with the right approach, they can be tackled with ease. In this article, we provided a Q&A guide to help you understand and solve radical equations. By following the steps outlined in this guide, you will be able to solve a wide range of radical equations and become more confident in your ability to tackle complex mathematical problems.

Additional Resources

If you are struggling to solve radical equations, there are many additional resources available to help you. Some of these resources include:

  • Online tutorials and videos
  • Math textbooks and workbooks
  • Online math communities and forums
  • Math tutors and mentors

By taking advantage of these resources, you will be able to improve your skills and become more confident in your ability to solve radical equations.

Final Tips

  • Always read the problem carefully and make sure you understand what is being asked.
  • Use a systematic approach to solving the equation, such as isolating the radical expression and then squaring both sides of the equation.
  • Check the solutions to make sure they satisfy the original equation.
  • Don't be afraid to ask for help if you are struggling to solve the equation.

By following these tips, you will be able to solve radical equations with ease and become more confident in your ability to tackle complex mathematical problems.