Question

If the roots of equation y² - √2y + √3 = 0  are α and β then α4 + β4  will be equal to 

1. 16 + 8√3 
2. 10 - 8√3 
3. 13 - 12√2 
4. 13 + 12√2 

Answer 1

Correct Answer - Option 2 : 10 - 8√3 

Concept:

For the quadratic equation ax² + bx + c = 0, (a ≠ 0), 

α + β = -b/a

αβ = c/a

Formula used:

α⁴ + β⁴ = (α² + β²)² - 2α²β² 

α2 + β2 = (α + β)² - 2αβ

(a - b)2 = a2 - 2ab + b2

Calculation:

y2 - √2y + √3 = 0     ---(1)

⇒ α + β = √2

αβ = √3

∵ α4 + β4 = (α2 + β2)2 - 2α2β2

⇒ α4 + β4 = [(α + β)2 - 2αβ]2 - 2α2β2

⇒ α4 + β4 = [(√2)2 - 2.√3]2 - 2(√3)²

⇒ α4 + β4 = (2 - 2√3)² - 6

∵ (a - b)² = a² - 2ab + b²

⇒  α4 + β4 = 4 - 8√3 + 12 - 6 

⇒  α4 + β4 = 10 - 8√3