Solving the Radical Equation: √x + √x-4 = 4 ! Solving Systems of Nonlinear Equations! Solving the Exponential Equation , Crack This Japanese Math Olympiad Problem: Are You Up for the Challenge?

Radical equations, those that involve square roots, can be intimidating at first glance. However, with a systematic approach, they become manageable and even enjoyable to solve. In this blog post, we’ll walk through the steps to solve the equation √x + √x-4 = 4  and unlock the mystery behind it.

 

Step 1: Isolate One of the Radicals

The equation √x + √x-4 = 4  contains two square roots. To solve it, we’ll start by isolating one of the radicals on one side of the equation. Let’s subtract from both sides:

√x = 4 -√x-4     

 

Step 2: Square Both Sides

To eliminate the square root, we’ll square both sides of the equation. Squaring the left side gives us X and squaring the right side requires us to expand:

x =(4 - √x-4)^2

 Expanding the right side:

x = 16 - 8√x-4 + (x – 4)  

Simplifying further:

x = x + 12 - 8√x-4   

Now, subtract x from both sides:

12 - 8√x-4 = 0  

 

Step 3: Solve for the Remaining Radical

 To isolate the remaining radical, subtract 12 from both sides:

-12 = - 8√x-4   

Next, divide both sides by -8:

12/8 = √x-4   

Simplify the fraction:

3/2 = √x-4   

 

Step 4: Square Both Sides Again

Square both sides to eliminate the square root:

(3/2)^2 = x - 4   

9/4 = x-4   

 

 Step 5: Solve for x

Finally, add 4 to both sides to solve for x:

x = 9/4 + 4   

Convert 4 into a fraction with the same denominator:

x = (9 + 16) /4 = 25/4

x = 6.25

 

Final Thoughts

Understanding and solving radical equations like √x + √x-4 = 4is an essential skill in algebra. By isolating radicals and carefully squaring both sides, you can untangle even the most complex-looking problems. Keep practicing, and soon, solving these equations will become second nature!

 GeT The Solution Here