Solve for x

Problem

√(,5*x+11)−√(,3*x+10)=1

Solution

1. Isolate one of the radical terms by adding √(,3*x+10) to both sides of the equation.

√(,5*x+11)=1+√(,3*x+10)

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

(√(,5*x+11))2=(1+√(,3*x+10))2

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

5*x+11=1+2√(,3*x+10)+(3*x+10)

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

5*x+11=3*x+11+2√(,3*x+10)

5. Isolate the remaining radical term by subtracting 3*x and 11 from both sides.

2*x=2√(,3*x+10)

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

x=√(,3*x+10)

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

x2=(√(,3*x+10))2

x2=3*x+10

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

x2−3*x−10=0

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

(x−5)*(x+2)=0

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

x=5

x=−2

11. Check for extraneous solutions by substituting the values back into the original equation. For x=5 √(,36)−√(,25)=6−5=1 (Valid). For x=−2 √(,1)−√(,4)=1−2=−1≠1 (Extraneous).

Final Answer

√(,5*x+11)−√(,3*x+10)=1⇒x=5

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