If x² + y² = 34 and xy = $10\dfrac{1}{2}$, find the value of 2(x + y)² + (x - y)².

We know that,

$2(x + y)^2 + (x - y)^2 = 2(x^2 + y^2 + 2xy) + (x^2 + y^2 - 2xy) \\[1em] = 2x^2 + 2y^2 + 4xy + x^2 + y^2 - 2xy \\[1em] = 3x^2 + 3y^2 + 2xy \\[1em] = 3(x^2 + y^2) + 2xy.$

Substituting values in above equation,

$= 3 \times 34 + 2 \times 10\dfrac{1}{2} \\[1em] = 102 + 2 \times \dfrac{21}{2} \\[1em] = 102 + 21 \\[1em] = 123.$

Hence, 2(x + y)² + (x - y)² = 123.