find the zeros of the quadratic polynomial and verify the relationship x² + 2√5x-15
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Let f(x) = 3x2 + 5x – 2 By splitting the middle term, we get f(x) = 3x2 + (6 – 1)x – 2 [∵ 5 = 6 – 1 and 2×3 = 6] = 3x2 + 6x – x – 2 = 3x(x + 2) – 1(x + 2) = (3x – 1) (x + 2) On putting f(x) = 0 , we get (3x – 1) (x + 2) = 0 ⇒ 3x – 1 = 0 or x + 2 = 0 x = 1/3 or x = – 2 Thus, the zeroes of the given polynomial 3x2 + 5x – 2 are – 2 and 1/3. Verification So, the relationship between the zeroes and the coefficients is verified.
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Given polynomial is x2−5
Let the zeros be α,β. (α>β)
x2−5=0⇒x2=5⇒x=±5
α=5,β=−5
So, α+β=0,αβ=−5
The coefficient of x is zero ⇒α+β=0
(& of x2 is 1).
Constant = αβ=−5
Hence verified.
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