How do you simplify #sqrt(x-1) + sqrt( 2x) = 3#?

Question

Eva Adamson · Accepted Answer

#rarrx=2# #rarrsqrt(x-1)+sqrt(2x)=3# #rarrsqrt(x-1)=3-sqrt(2x)# #rarr[sqrt(x-1)]^2=[3-sqrt(2x)]^2# #rarrx-1=9-6sqrt(2x)+2x# #rarr6sqrt(2x)=x+10# #rarr[6sqrt(2x)]^2=[x+10]^2# #rarr36*(2x)=x^2+20x+100# #rarrx^2-52x+100=0# #rarrx^2-2*x*26+26^2-26^2+100=0# #rarr(x-26)^2=26^2-100=576# #rarrx-26=sqrt(576)=+-24# #rarrx=26+24,26-24=50 or 2# Putting #x=50# in given equation, we get, #rarrsqrt(50-1)+sqrt(2*50)=17(rejected)# Putting #x=2# in given equation, we get, #rarrsqrt(2-1)+sqrt(2*2)=3(accepted)# So, the required value of x is #2.#

Hazel Anglin · Answer

undefined To simplify the equation sqrt(x-1) + sqrt(2x) = 3, you can follow these steps:

1. Isolate one of the square root terms on one side of the equation. Let's isolate sqrt(x-1) by subtracting sqrt(2x) from both sides:
   sqrt(x-1) = 3 - sqrt(2x)

2. Square both sides of the equation to eliminate the square root:
   (sqrt(x-1))^2 = (3 - sqrt(2x))^2
   x - 1 = 9 - 6sqrt(2x) + 2x

3. Simplify the equation by combining like terms:
   x - 1 = 9 + 2x - 6sqrt(2x)

4. Move all terms involving x to one side of the equation and all constant terms to the other side:
   x - 2x = 9 + 1 - 6sqrt(2x)
   -x = 10 - 6sqrt(2x)

5. Square both sides of the equation again to eliminate the square root:
   (-x)^2 = (10 - 6sqrt(2x))^2
   x^2 = 100 - 120sqrt(2x) + 36(2x)
   x^2 = 100 - 120sqrt(2x) + 72x

6. Rearrange the equation to bring all terms to one side:
   x^2 - 72x - 120sqrt(2x) + 100 = 0

This is the simplified form of the equation sqrt(x-1) + sqrt(2x) = 3.