If the roots of the quadratic equation x2 – x + 1 = 0 are α and β, then find the quadratic equation whose roots are α4 and β4.
If the roots of the quadratic equation x2 – x + 1 = 0 are α and β, then find the quadratic equation whose roots are α4 and β4. Correct Answer x<sup style="">2</sup> + x + 1 = 0
Given:
The roots of the quadratic equation X2 – x + 1 = 0 are α and β.
Concept used:
If two roots are given, then the quadratic equation will be x2 – (sum of roots)x + product of roots = 0
Calculation:
The roots of the quadratic equation x2 – x + 1 = 0 are α and β.
⇒ sum of roots will be α + β = 1 and product of roots will be α × β = 1
α × β = 1 ⇒ α = 1/β
⇒ α + β = α + 1/α = 1
α4 + β4 = α4 + 1/α4
α + 1/α = 1
⇒ (α + 1/α)2 = 1
⇒ α2 + 1/α2 + 2 × α × 1/α = 1
⇒ α2 + 1/α2 = 1 – 2 = -1
⇒ (α2 + 1/α2)2 = (-1)2
⇒ α4 + 1/α4 + 2 × α2 × 1/α2 = 1
⇒ α4 + 1/α4 = 1 – 2 = -1
α × β = 1 ⇒ α4.β4 = 1
The quadratic equation whose roots are α4 and β4 will be x2 – (α4 + β4)x + α4.β4 = 0.
x2 – (α4 + β4)x + α4.β4 = x2 + x + 1 = 0
∴ The quadratic equation whose roots are α4 and β4 will be x2 + x + 1 = 0