Solve for x

Problem

√(,3*x+1)−√(,x−1)=2

Solution

1. Isolate one of the radical terms by adding √(,x−1) to both sides of the equation.

√(,3*x+1)=√(,x−1)+2

2. Square both sides of the equation to eliminate the radical on the left side.

(√(,3*x+1))2=(√(,x−1)+2)2

3. Expand the right side using the identity (a+b)2=a2+2*a*b+b2

3*x+1=(x−1)+4√(,x−1)+4

4. Simplify the equation by combining like terms on the right side.

3*x+1=x+3+4√(,x−1)

5. Isolate the remaining radical term by subtracting x and 3 from both sides.

2*x−2=4√(,x−1)

6. Divide both sides by 2 to simplify the coefficients before squaring again.

x−1=2√(,x−1)

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

(x−1)2=(2√(,x−1))2

8. Expand and simplify both sides of the resulting equation.

x2−2*x+1=4*(x−1)

x2−2*x+1=4*x−4

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

x2−6*x+5=0

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

(x−5)*(x−1)=0

11. Solve for x by setting each factor equal to zero.

x=5

x=1

12. Check the solutions in the original equation. For x=5 √(,16)−√(,4)=4−2=2 (True). For x=1 √(,4)−√(,0)=2−0=2 (True).

Final Answer

√(,3*x+1)−√(,x−1)=2⇒x=1,5

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