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If both roots of ax^2+bx+c are more than 2 then prove that the value of a^2+c^2+2ac-b^2 is always positive

If both roots of ax^2+bx+c are more than 2 then prove that the value of a^2+c^2+2ac-b^2 is always positive

Grade:11

2 Answers

Arun
25750 Points
5 years ago
Dear student
 
If both the roots are greater than 2 then
By sum and product of roots
-b/a > 4
- b> 4a 
And
c/a >4
c> 4a
Now
a² +c² + 2ac - b² = (a+c)² - b² = (5a)² - (4a)² = 9a² which is always positive
Asad Rehman
27 Points
5 years ago
α and β are the root of the equationax^2 +bx + c =0 , so , α + β = -b/a and αβ =c/aNow, -1/α +(-1/β) = -{(-b/a)/c/a}=b/cand (-1/α)(-1/β) = a/c . Hence the required quadratic equation isx^2 -(sum of the roots)*x+product of the root=0 i.e. x^2 - (b/c)x + a/c = 0i.e. cx^2 -bx +a =0

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