If x = √(2)+1 and y= √(2) over 3 , then what is the value of (x-2y-1)(x²+4y²+2xy-2x-2y+1) ?

If x = √(2)+1 and y= √(2) over 3 , then what is the value…

If $x = { \sqrt{2}}+1$ and $y= {\sqrt{2} \over 3}$ , then what is the value of ${{(x-2y-1)(x^2+4y^2+2xy-2x-2y+1)}}$ ?

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Correct answer: A

Explanation

The correct answer is Option 1 i.e. ${38\sqrt{2} \over 27}$ .

a³ - b³ = (a - b)(a² + ab + b²)

(a - b)² = a² - 2ab + b³

Rearranging the given expression,

(x - 2y - 1)((x² - 2x + 1) + 4y² + (2xy - 2y))

⇒ (x - 2y - 1)((x - 1)² + 4y² + 2y(x - 1)) = (x - 1)³ - 2y³

Putting values of x and y,

⇒ $(\sqrt{2} + 1)^3$ - 2 $(\frac{\sqrt{2}}{3})^3$

⇒ $2\sqrt{2}$ - 4 $\frac{\sqrt{2}}{27}$ = ${38\sqrt{2} \over 27}$