Solving the Radical Equation:  √x + √(x-4) =4 - Step-by-Step Guide! Solving A Nonlinear Equation

Square root equations can be a little tricky, but they’re also incredibly rewarding to solve once you get the hang of them. In this post, we’ll dive into a specific square root equation √x + √(x-4) =4. If you’ve ever found yourself confused by equations involving square roots, this step-by-step breakdown will make solving them a breeze. Let’s walk through the process and solve this mind-boggling algebra problem together!

 

The Equation: √x + √(x-4) =4

This equation involves two square roots, so the goal is to eliminate them by squaring both sides of the equation. But before we do that, we need to isolate the square roots and simplify the problem.

 

Step 1: Isolate One Square Root

First, let’s isolate one of the square roots. We’ll start by moving √(x-4) to the other side of the equation:

√x =4 - √(x-4)

 

Step 2: Square Both Sides

Now that we’ve isolated one square root, we can square both sides of the equation to eliminate it. Squaring both sides gives:

(√x)^2 = {4 - √x-4}^2

Simplifying both sides:

X = {4 - √x-4}^2

On the right-hand side, apply the square of a binomial formula:

X = 16 – 8√x-4 + x-4

Simplify the expression:

X = 12 – 8√x-4 + x

 

Step 3: Eliminate the Square Root

Subtract x from both sides:

0 = 12 - 8√x-4

Move 12 to the other side:

-12 = -8√x-4

Now divide both sides by -8 to isolate the square root:

12/8 = √x-4

Simplify:

3/2 = √x-4

 

Step 4: Square Both Sides Again

To eliminate the square root, square both sides:

(3/2)^2 = x-4

This simplifies to:

9/4 = x -4

 

Step 5: Solve for x

Now, solve for x by adding 4 to both sides:

X = 4 + 9/4

To add these, convert 4 into a fraction:

X = (16+9)/ 4

X= 25/4

Step 6: Verify the Solution

Finally, let’s verify the solution by substituting x = 25/4 =6.25 back into the original equation:

√{6.25} + √sqrt{6.25 - 4} = 4

Simplify:

2.5 + √{2.25} = 4

2.5 + 1.5 = 4

Since both sides are equal, the solution is correct!

 

Conclusion: Understanding Square Root Equations

The equation √{x} + √{x-4} = 4 may look complex at first, but breaking it down into manageable steps makes it much easier to solve. By isolating the square roots, squaring both sides, and carefully simplifying, we found that x = 6.25 is the solution. Mastering square root equations like this one will improve your algebra skills and prepare you for more challenging problems.

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