Solving √(2x - 9) = √(9 - 2x): A Step-by-Step Guide

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Hey guys! Let's dive into solving this radical equation: √(2x - 9) = √(9 - 2x). Radical equations might seem intimidating at first, but with a systematic approach, they can be conquered. In this comprehensive guide, we'll break down each step, ensuring you grasp the underlying concepts and can confidently tackle similar problems. So, grab your thinking caps, and let's get started!

Understanding the Basics of Radical Equations

Before we jump into the solution, let's quickly recap what radical equations are and the key principles involved in solving them. Radical equations are equations where the variable appears inside a radical, most commonly a square root. To solve them, our primary goal is to isolate the radical term and then eliminate it by raising both sides of the equation to the appropriate power. This process often involves algebraic manipulation and careful attention to potential extraneous solutions.

When dealing with square roots, we square both sides. For cube roots, we cube both sides, and so on. This eliminates the radical, allowing us to solve for the variable. However, it's crucial to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, always check your solutions in the original equation.

Step-by-Step Solution

Now, let's tackle the equation √(2x - 9) = √(9 - 2x) step by step.

1. The Initial Equation

We start with the given equation:

√(2x - 9) = √(9 - 2x)

2. Squaring Both Sides

To eliminate the square roots, we square both sides of the equation. This is a crucial step in solving radical equations.

(√(2x - 9))^2 = (√(9 - 2x))^2

This simplifies to:

2x - 9 = 9 - 2x

3. Isolating the Variable

Next, we want to isolate the variable x. To do this, we'll add 2x to both sides of the equation:

2x - 9 + 2x = 9 - 2x + 2x

This gives us:

4x - 9 = 9

Now, add 9 to both sides:

4x - 9 + 9 = 9 + 9

Which simplifies to:

4x = 18

4. Solving for x

To solve for x, we divide both sides by 4:

4x / 4 = 18 / 4

So,

x = 18 / 4

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

x = 9 / 2

Thus, our potential solution is x = 9/2.

5. Checking for Extraneous Solutions

Remember, it's absolutely essential to check our solution in the original equation to ensure it's not extraneous. Let's substitute x = 9/2 back into the original equation:

√(2(9/2) - 9) = √(9 - 2(9/2))

Simplify inside the square roots:

√(9 - 9) = √(9 - 9)

√(0) = √(0)

0 = 0

Since the equation holds true, x = 9/2 is a valid solution.

Final Answer

Therefore, the solution to the equation √(2x - 9) = √(9 - 2x) is:

x = 9/2

Common Mistakes to Avoid

When solving radical equations, it's easy to make a few common mistakes. Being aware of these pitfalls can save you a lot of headaches.

Forgetting to Check for Extraneous Solutions

As we've emphasized, this is the most crucial step. Squaring both sides can introduce solutions that don't work in the original equation. Always plug your solutions back into the original equation to verify them.

Incorrectly Squaring Binomials

If you have an equation like √(x + 2) = x, squaring both sides requires careful attention. Remember that (x + 2)^2 is not x^2 + 2^2. It's (x + 2)(x + 2) = x^2 + 4x + 4. Make sure to expand binomials correctly.

Not Isolating the Radical First

Before squaring, isolate the radical term. For example, if you have √(x) + 3 = x, subtract 3 from both sides first to get √(x) = x - 3. Squaring before isolating can lead to more complicated algebra.

Practice Problems

To solidify your understanding, let's look at a few more examples.

Example 1

Solve: √(3x + 4) = 5

  1. Square both sides: (√(3x + 4))^2 = 5^2

    3x + 4 = 25

  2. Isolate x: 3x = 21

  3. Solve for x: x = 7

  4. Check: √(3(7) + 4) = √(25) = 5. Solution is valid.

Example 2

Solve: √(x - 1) = x - 3

  1. Square both sides: (√(x - 1))^2 = (x - 3)^2

    x - 1 = x^2 - 6x + 9

  2. Rearrange: 0 = x^2 - 7x + 10

  3. Factor: 0 = (x - 2)(x - 5)

  4. Solve for x: x = 2, x = 5

  5. Check:

    • For x = 2: √(2 - 1) = √(1) = 1, 2 - 3 = -1. Extraneous.
    • For x = 5: √(5 - 1) = √(4) = 2, 5 - 3 = 2. Valid.

So, the only solution is x = 5.

Conclusion

Solving radical equations requires a systematic approach: isolate the radical, eliminate it by raising both sides to the appropriate power, solve for the variable, and always check for extraneous solutions. By following these steps and avoiding common mistakes, you'll be well-equipped to tackle a wide range of radical equations. Remember, practice makes perfect, so keep working on problems, and you'll become a pro in no time! Happy solving, guys!