Roots lie in an interval:
Here normally we will discuss different useful and sufficient conditions we could impose on a quadratic equation ax2 + bx + c = 0 so that roots of the provided equation lies in a subsequent interval. Since a ≠ 0 , we may take
Case I: Both the roots are positive i.e. they lie in (0, ∞), then the sum of the roots as well as the multiplication of the roots have to be positive.
Case II: Both the roots are negative i.e. they lie in (-∞, 0), then the addition of the roots have to be negative and the product of the roots have to be positive.
Case III: One root is negative and other is positive i.e. origin is lying between the roots. Normally f(0)<0 is the required and sufficient condition.
Case IV: Both the roots are bigger then a real number k.
D ≥ 0 ... (1)
f(k) > 0 ...(2)
-b/2a > k ...(3)
These are required & sufficient conditions.
Case V: If both the roots are less than a real number k.
D ³ 0 ... (1)
f(k) > 0 ...(2)
-b/2a< k ...(3)
These are required & sufficient conditions.
Case VI: A real number k is defining between the roots i.e. one root is less then k and other is larger than k.
D > 0 ... (1)
f(k) < 0 ... (2)
These require & sufficient conditions.
Case VII: exactly are root is relying between k1 and k2
f(k1) < 0 and f(k2) > 0 f(k1) > 0 and f(k2) < 0
Hence the sufficient condition is f(k1). f(k2) < 0
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