Quadratic Expressions

Quadratic Expressions

Let F(x) = ax ² + bx + c, a,b,c ∈ R, a ≠ 0.

The graph of f (x) is a parabola with vertex at .

The mouth of the parabola opens up or down according as 'a' is positive or negative respectively.

Three cases arise

(i) If D < 0, aF(x) > 0 for all values of x.

(ii) If D = 0, F(x) = 0 has two equal roots and aF(x) ≥ 0.

(iii) If D > 0, F(x) = 0 has two real and distinct roots say α and β. Let α< β.

In this case a F(x) < 0 for all x ∈ (α, β), a F(α) = 0, a F(β) = 0 and
a F(x) > 0 ∀ x ∈ (−∞, α) ∪ (β, ∞).

(iv) If D ≥ 0, a F(k) > 0 for any real number k and - > k. Then both the roots of
F(x) = 0 are greater than k.

(v) If D ≥ 0, a F(k) > 0 and − < k for any real number k, then both the roots are smaller than k.

(vi) If D > 0 and a F(k) < 0 for any real number k, then k lies between the roots of
F(x) = 0.

(vii) Exactly one root of F(x) = 0 lies between the real numbers k1 and k2 if
F(k1) F(k ₂ ) < 0.