Solve for x square root of x+ square root of 2x=1

Problem

√(,x)+√(,2*x)=1

Solution

1. Factor out the common term √(,x) from the left side of the equation.

√(,x)*(1+√(,2))=1

2. Isolate the square root term by dividing both sides by (1+√(,2))

√(,x)=1/(1+√(,2))

3. Square both sides of the equation to solve for x

x=(1/(1+√(,2)))^2

4. Expand the denominator using the square of a binomial formula (a+b)^2=a^2+2*a*b+b^2

x=1/(1+2√(,2)+2)

5. Simplify the denominator by combining like terms.

x=1/(3+2√(,2))

6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 3−2√(,2)

x=(3−2√(,2))/((3+2√(,2))*(3−2√(,2)))

7. Simplify the denominator using the difference of squares (a+b)*(a−b)=a^2−b^2

x=(3−2√(,2))/(9−8)

8. Finalize the value of x

x=3−2√(,2)

Final Answer

√(,x)+√(,2*x)=1⇒x=3−2√(,2)

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