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Question
Find the condition that one of the roots of ax2 + bx + c may be negative of the other
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Solution
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 one root is the negative of the other
β = – α
(2) ⇒ α + (– α) = `- "b"/a"`
0 = `- "b"/a"`
⇒ b = 0
(3) ⇒ α(– α) = `"c"/"a"`
– α2 = `"c"/"a"`
Hence the required condition is b = 0
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