Solve for x

Problem

√(,4*x+1)−√(,x−2)=3

Solution

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

√(,4*x+1)=3+√(,x−2)

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

(√(,4*x+1))2=(3+√(,x−2))2

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

4*x+1=9+6√(,x−2)+(x−2)

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

4*x+1=x+7+6√(,x−2)

5. Isolate the remaining radical term by subtracting x and 7 from both sides.

3*x−6=6√(,x−2)

6. Divide both sides by 3 to simplify the coefficients before squaring again.

x−2=2√(,x−2)

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

(x−2)2=(2√(,x−2))2

8. Expand and simplify the resulting quadratic equation.

x2−4*x+4=4*(x−2)

x2−4*x+4=4*x−8

9. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by moving all terms to one side.

x2−8*x+12=0

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

(x−2)*(x−6)=0

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

x=2

x=6

12. Verify the solutions by substituting them back into the original equation. For x=2 √(,9)−√(,0)=3 which is 3=3 For x=6 √(,25)−√(,4)=3 which is 5−2=3 Both are valid.

Final Answer

√(,4*x+1)−√(,x−2)=3⇒x=2,6

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