Solve for x

Problem

√(,7*x+1)−√(,x−5)=6

Solution

1. Isolate one of the radical terms by adding √(,x−5) to both sides of the equation.

√(,7*x+1)=6+√(,x−5)

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

(√(,7*x+1))2=(6+√(,x−5))2

7*x+1=36+12√(,x−5)+(x−5)

3. Simplify the right side by combining like terms.

7*x+1=31+x+12√(,x−5)

4. Isolate the remaining radical term by subtracting x and 31 from both sides.

6*x−30=12√(,x−5)

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

x−5=2√(,x−5)

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

(x−5)2=(2√(,x−5))2

x2−10*x+25=4*(x−5)

7. Expand and move all terms to one side to form a quadratic equation.

x2−10*x+25=4*x−20

x2−14*x+45=0

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

(x−9)*(x−5)=0

x=9,x=5

9. Check the solutions in the original equation to ensure they are not extraneous.

For *x=9:√(,7*(9)+1)−√(,9−5)=√(,64)−√(,4)=8−2=6

For *x=5:√(,7*(5)+1)−√(,5−5)=√(,36)−√(,0)=6−0=6

Final Answer

√(,7*x+1)−√(,x−5)=6⇒x=5,9

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