Question

ax² + bx + c = 0 have Common roots. If one root is p + iq .Then other root is p - iq.
Prove it.


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Answer 1

Question:-

☞ ax² + bx + c = 0 have Common roots. If one root is p + iq .Then other root is p - iq.

Prove it.

☞ Proof:

☞ Given ax² + bx + c = 0

let one root is p + √q => α

let other root is β

α + β = -b/a

=> β = -b/a - p -√q .....(1)

αβ = c/a

B = c/aα

β = c ......(2)

a(p+√q)

☞ (1) = (2)

-b/a -p -√q = c

a(p+√q)

-b - (p + √q)a = c

a a (p + √q)

=> -b (p+√q) - (p+√q)²a = c

=> c + b (p+√q) + (p+√q)²a = 0

=> c +bp + b√q + p²a + qa + 2p√q a = 0

=> (c +bp + p²a + qa) + (b√q + 2p√q a) = 0

=> c +bp + p²a + qa =0

b√q + 2p√q a =0 => √q ( b + 2pa) = 0

=> b + 2pq = 0 [ q ≠ 0 ]

=> 2pa = -b

=> 2p = -b/a

(1) ==> β = -b/a - p -√q

= 2p - p - √q

=> β = p - √q

☞ Hence proved.