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If $α$ and $β$ are the zeros of the quadratic polynomial $f (x) = a x 2 + b x + c$ , then evaluate $1 α − 1 β .$

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b 2 − 4 a c c

− b 2 − 4 a c c

− b 2 + 4 b c c

− b 2 − 4 a c b

answer is A.

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Detailed Solution

Given that

α

and

β

are the zeroes of the quadratic polynomial

f (x) = a x 2 + b x + c .

Then sum and product of roots can be written as

Sum of zeroes = − b a ⇒ α + β = − b a

And,Product = c a ⇒ α β = c a

Given to evaluate 1 α − 1 β,

Use the identity,

a − b 2 = a + b 2 − 4 a b ⇒ a − b = (a + b) 2 − 4 a b

Therefore,

1 α − 1 β

can be written as

1 α + 1 β 2 − 4 1 α × 1 β

1 α − 1 β = α + β α β 2 − 4 α β

Substitute

α + β = − b a and α β = c a

into the equation,

1 α − 1 β = α + β α β 2 − 4 α β ⇒ 1 α − 1 β = − b a c a 2 − 4 c a ⇒ 1 α − 1 β = − b a × a c 2 − 4 a c

⇒ 1 α − 1 β = b 2 c 2 − 4 a c ⇒ 1 α − 1 β = b 2 − 4 a c c 2 ⇒ 1 α − 1 β = b 2 − 4 a c c

Therefore, value of

1 α − 1 β is b 2 − 4 a c c

.
Hence the correct option is 1.

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