If α and β are the roots of the equation x² – 7x + 11 = 0, then the value of α³ + β³ is equal to :
112
77
49
224
Answer is Right!
Answer is Wrong!
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UPSC CISF-AC-EXE – 2022
Let the roots be α and β.
According to Vieta’s formulas, for a quadratic equation ax² + bx + c = 0, the sum of the roots is α + β = -b/a and the product of the roots is αβ = c/a.
For the given equation x² – 7x + 11 = 0 (where a=1, b=-7, c=11):
Sum of roots: α + β = -(-7)/1 = 7.
Product of roots: αβ = 11/1 = 11.
We need to find the value of α³ + β³.
We can use the algebraic identity for the sum of cubes: α³ + β³ = (α + β)(α² – αβ + β²).
We can express α² + β² in terms of (α + β) and αβ:
α² + β² = (α + β)² – 2αβ.
Substituting this into the identity:
α³ + β³ = (α + β)[((α + β)² – 2αβ) – αβ]
α³ + β³ = (α + β)((α + β)² – 3αβ).
Now, substitute the values we found for (α + β) and αβ:
α + β = 7
αβ = 11
α³ + β³ = (7)((7)² – 3 × 11)
α³ + β³ = 7(49 – 33)
α³ + β³ = 7(16)
α³ + β³ = 112.