The sum of the roots of a quadratic equation is 2 and the sum of the cubes of the roots is 98. The equation is?
- x² - 4x + 15 = 0
- x² - 4x - 15 = 0
- x² - 2x - 15 = 0
- None of these
Explanation
Given:
1. Sum of roots: α + β = 2
2. Sum of cubes of roots: α³ + β³ = 98
Step-by-Step Solution
Step 1: Use the identity for α³ + β³
α³ + β³ = (α + β)³ - 3αβ(α + β)
Step 2: Substitute known values
98 = (2)³ - 3αβ(2)
98 = 8 - 6αβ
Step 3: Solve for αβ
98 - 8 = -6αβ
90 = -6αβ
αβ = -15
Step 4: Form the quadratic equation
x² - (α + β)x + αβ = 0
x² - 2x - 15 = 0
The answer is x² - 2x - 15 = 0.
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