Quadratic Polynomial

Algebra Level 3

Define a polynomial relation f ( x ) f(x) = = a x 2 + b x + c ax^2 + bx+c = 0 0 where a a is a positive real number. If the discriminant of f ( x ) f(x) is less than 0 0 ,need it always be true f ( x ) > 0 f(x)>0 ?

  • NOTE: Discriminant = b 2 − 4 a c b^2-4ac
No Yes

Aman Thegreat by Aman thegreat , 19, India

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1 solution

f ( x ) = a x 2 + b x + c = a ( x + b 2 a ) 2 − b 2 − 4 a c 4 a f(x)=ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a} . If a > 0 a\gt 0 and b 2 − 4 a c < 0 b^2-4ac\lt 0 , then f ( x ) > 0 f(x)\gt 0 ∀ x ∈ R \forall x\in\R .

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