Let ααα and ββ be the roots of x2−3x+c=0x2−3x+c=0, where cc is a real number. If −α−α is a root of x2+3x−c=0x2+3x−c=0, find the value of αβαβ.
Answer:
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- α & βα & β are the roots of the equation, therefore,
αβ=c1=c .....(1)αβ=c1=c .....(1) - As αα is the root of the equation x2−3x+c=0x2−3x+c=0,
α2−3α+c=0 .....(2)α2−3α+c=0 .....(2)
Also, −α−α is the root of the equation x2+3x−c=0x2+3x−c=0,
(−α)2+3(−α)−c=0⟹α2−3α−c=0.....(3) - On subtracting eq(2) by eq(3), we get,
α2−3α−c−(α2−3α+c)=0⟹α2−3α−c−α2+3α−c=0⟹−2c=0⟹c=0 - By eq(1), we have,
αβ=c⟹αβ=0