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प्रश्न
Find the condition that one of the roots of ax2 + bx + c may be thrice the other
उत्तर
The given quadratic equation is
ax2 + bx + c = 0 .....(1)
Let α and β be the roots of the equation (1) then
Sum of the roots α + β = 0 .....(2)
Product of the roots αβ = 0 .....(3)
Given that one root is thrice the other
β = 3α
(2) ⇒ α + 3α = `- "b"/"a"`
4α = `- "b"/"a"`
α = `- "b"/(4"a")`
(3) ⇒ α(3α) = `"c"/"a"`
3α2 = `"c"/"a"` ⇒ α2 = `"c"/(3"a")`
`(- "b"/(4"a"))^2 = "c"/(3"a')` ⇒ `"b"^2/(16"a"^2) = "c"/(3"a")`
`"b"^2/(16"a") = "c"/3`
3b2 = 16ac
When is the required condition?
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