Solve for x

Problem

2√(,x+1)−√(,2*x+3)=1

Solution

1. Isolate one of the radical terms by adding √(,2*x+3) to both sides of the equation.

2√(,x+1)=1+√(,2*x+3)

2. Square both sides of the equation to begin eliminating the radicals.

(2√(,x+1))2=(1+√(,2*x+3))2

3. Expand both sides, applying the identity (a+b)2=a2+2*a*b+b2 to the right side.

4*(x+1)=1+2√(,2*x+3)+(2*x+3)

4. Simplify the equation by combining like terms and isolating the remaining radical term.

4*x+4=2*x+4+2√(,2*x+3)

2*x=2√(,2*x+3)

5. Divide both sides by 2 to simplify the expression before squaring again.

x=√(,2*x+3)

6. Square both sides again to eliminate the final radical.

x2=2*x+3

7. Rearrange the equation into a standard quadratic form a*x2+b*x+c=0

x2−2*x−3=0

8. Factor the quadratic equation to find the potential values for x

(x−3)*(x+1)=0

x=3,x=−1

9. Check for extraneous solutions by substituting the values back into the original equation. For x=3 2√(,4)−√(,9)=4−3=1 (True). For x=−1 2√(,0)−√(,1)=0−1=−1 (False).

Final Answer

x=3

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