If alpha and beta are the zeros of a quadratic polynomial x^2 - px + q then find the value alpha^2 + beta^2 and 1/alpha + 1/beta
Answers
Answer.......
Note the important points.
- Zeros = −Coefficient of xCoefficient of x2
α + β = - p
- whole square on both hand side
( α + β )2 = p 2
α2 + β2 + 2 α β = p2⇒α2 + β2 + 2 α β −
2αβ + 2αβ= p2⇒α2 + β2 − 2 α β +4 αβ= p2
(α − β) 2 +4 αβ= p2
also.
- Products of zeros = Constant termCoefficient of x2
α β = q
- Substitute that value in equation 3 ,
- Then we get
(α− β)2 + 4 (q ) = p2
(α− β)2 +4 q= p2
(α− β)2 = p2 − 4 q −
Now adding equation 2 and 4
(α + β)2 + (α − β)2 = p2 + p2 − 4 q= 2 p2 − 4 q
And also multiply equation 2 and 4
(α + β)2 × (α − β)2 = p2( p2 − 4 q)= p4 − 4 p2q
((Formula of polynomial))
- Sum of zeros and product of zeros
- Polynomial = k [ x2 - ( Sum of zeros ) x + ( Product of zeros ) ] ,
- Therefore k is any non zero real number.
((Substituting values ))
- Quadratic polynomial = k [ x2 - ( 2 p2 - 4 q) x + ( 2 p4 - 4 p2q) ]
final answer to make...
= x2 - ( 2 p2 - 4 q) x + ( 2 p4 - 4 p2q) [ k = 1 ]
hence equation solved...
Answer:
Answer in the image
Step-by-step explanation:
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