The expression ax2 + bx + c is said to be a real quadratic expression in x where a, b, c are real and a ≠ 0.

Let f(x) = ax2 + bx + c  where a, b, c, ∈ R (a ≠ 0). Now  f(x) can be rewritten as  f(x) = = a……….(1), where  D = b2 – 4ac is the discriminant of the quadratic expression.  From (1) it is clear that f(x) = ax2 + bx + c will represent a parabola  whose axis is parallel to the y-axis, and vertex is at A.

It is also clear that if a > 0, the parabola  will be open upward and if a < 0 the parabola  will be open downward and it depends on the sign of b2 –4ac  that the parabola  cuts the x-axis at two points ( b2-4ac > 0), touches the  x-axis (b2- 4ac = 0) or never intersects with the x-axis(b2-4ac < 0). Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah.

If a > 0

Sub case A: a  > 0 and b2 - 4ac < 0 ⇔ f(x) > 0 ∀ x ∈ R.

In this case the parabola always remains open upward and above the x-axis.

Sub case B:  a > 0 and b2 – 4ac = 0  ⇔  f(x) ≥ 0  ∀ x  ∈ R.

In this case the parabola touches the x-axis at one point and remains open upward.

Sub case C:   a > 0 and b2 - 4ac > 0. Let f(x) = 0 has two real roots α and β (α < β). Then f(x) > 0 ∀  x ∈ (-∞, α)∪(β, ∞)and f(x) < 0  ∀ x∈ (α, β)

In this case the parabola cuts the x- axis at two points α and β and remains open upward.

Greatest and least value of a quadratic expression ax2 + bx + c when a>0:

In this case ax2 + bx + c has no greatest value and it has least value at x = – .

Case II:  If a > 0

Sub case A: a < 0 and b2 - 4ac < 0⇔ f(x) < 0   ∀  x ∈ R.

In this case the parabola remains open downward and always below the x-axis.

Sub case B:   a < 0  and b2 – 4ac  = 0⇔    f(x) ≤ 0  ∀  x ∈ R.

In this case the parabola touches the x - axis and remains open downward.

Sub case C: a < 0 and b2 - 4ac > 0.

Let f(x) = 0 have two real roots α and β (α < β).Then  f(x) < 0 ∀  x ∈  (-∞, α)∪(β, ∞) and  f(x) > 0 ∀  x ∈  (α, β).

Greatest and least value of a quadratic expression ax2 + bx + c when a<0:

If a < 0, then ax2 + bx + c has no least value and it has greatest value at x = – .