Solve for x

Problem

√(,16+x)+√(,16−x)=8

Solution

1. Isolate one of the radical terms by subtracting √(,16−x) from both sides of the equation.

√(,16+x)=8−√(,16−x)

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

(√(,16+x))2=(8−√(,16−x))2

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

16+x=64−16√(,16−x)+(16−x)

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

16+x=80−x−16√(,16−x)

5. Isolate the remaining radical term by moving all other terms to the left side.

2*x−64=−16√(,16−x)

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

−x+32=8√(,16−x)

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

(−x+32)2=(8√(,16−x))2

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

x2−64*x+1024=64*(16−x)

x2−64*x+1024=1024−64*x

9. Solve for x by canceling identical terms on both sides.

x2=0

x=0

10. Verify the solution by substituting x=0 back into the original equation.

√(,16+0)+√(,16−0)=4+4=8

Final Answer

√(,16+x)+√(,16−x)=8⇒x=0

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