If the sum of the roots of the quadratic equation ax square+ bx+c=0
is equal to the sum of the squares of their reciprocals, then
prove that 2ca sq. = bc sq.+ ab sq
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2ca² = bc² + ab² If the sum of roots = sum of squares of their reciprocals
Step-by-step explanation:
ax² + bx + c = 0
let say α , β are the roots
then
α + β = 1/α² + 1/β²
LHS =
α + β
= - b/a
RHS
= 1/α² + 1/β²
= (β² + α²)/α²β²
= ((α + β)² - 2αβ)/α²β²
= ((α + β)² - 2αβ)/(αβ)²
αβ = c/a
= ( (-b/a)² - 2(c/a) ) /(c/a)²
= (b² - 2ac)/c²
- b/a = (b² - 2ac)/c²
=> -bc² = ab² - 2a²c
=> 2a²c = ab² + bc²
=> 2ca² = bc² + ab²
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