Question

If α and β are the distinct roots of the equation `x^2 + (3)^(1/4)x + 3^(1/2)` = 0, then the value of α⁹⁶(α¹² – 1) + β⁹⁶(β¹² – 1) is equal to ______.

Options

• 56 × 3²⁵

• 52 × 3²⁴

• 56 × 3²⁴

• 28 × 3²⁵

Answer 1

If α and β are the distinct roots of the equation `x^2 + (3)^(1/4)x + 3^(1/2)` = 0, then the value of α⁹⁶(α¹² – 1) + β⁹⁶(β¹² – 1) is equal to `underlinebb(52 xx 3^24)`.

Explanation:

Given `x^2 + (3)^(1/4)x + 3^(1/2)` = 0

⇒ `x^2 + sqrt(3) = -3^(1/4)x`

Squaring both sides,

⇒ `x^4 + 2sqrt(3)x^2 + 3 = sqrt(3)x^2`

⇒ `x^4 + sqrt(3)x^2 + 3` = 0

⇒ x⁴ + 3 = `-sqrt(3)x^2`

Now squaring both the sides again,

⇒ x⁸ + 6x⁴ + 9 = 3x⁴

⇒ x⁸ + 3x⁴ + 9 = 0

Put x = α, α⁸ = –9α – 3α⁴

∴ α¹² = –9α⁴ – 3α⁸ = –9α⁴ –3(–9 – 3α⁴) = 27

Similarly β¹² = 27

⇒ α⁹⁶(α¹² – 1) + β⁹⁶(β¹² – 1)

= (27)⁸ × 26 + (27)⁸ × 26 = 52 × (27)⁸

= 52 × 3²⁴