Quadratic expression:
The expression ax2 + bx + c is called 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) may be written as f(x) =
.........(1), where D = b2 - 4ac is the discriminant of the quadratic expression. From (1) it is defined that f(x) = ax2 + bx + c may show a parabola whose axis is parallel to the y-axis, and vertex is at
.
It is also obvious that if a > 0, the parabola can be open upward and if a < 0 the parabola can be open downward and it relays on the sign of b2 -4ac that the parabola intersects the x-axis at two points ( b2-4ac > 0), contacts the x-axis (b2- 4ac = 0) or never cuts with the x-axis (b2-4ac < 0).
Case I: If a > 0
Sub case A: a>0 and b2 - 4ac < 0 ⇔ f(x) > 0 ∀ x ∈ R.
In that case the parabola usually remains open upward and above the x-axis.
![1506_Quadratic expression2.png](https://www.expertsmind.com/../CMSImages/1506_Quadratic expression2.png)
Sub case B: a > 0 and b2 - 4ac = 0 ⇔ f(x) ≥ 0 ∀ x ∈ R.
In that type the parabola contact with the x-axis at one point and stays open upward.
![484_Quadratic expression3.png](https://www.expertsmind.com/../CMSImages/484_Quadratic expression3.png)
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 that case the parabola intersect the x- axis at two points a and b and stays open upward.
![485_Quadratic expression4.png](https://www.expertsmind.com/../CMSImages/485_Quadratic expression4.png)
Greatest and least value of a quadratic expression ax2 + bx + c when a>0:
In that case ax2 + bx + c has no largest value and it has least value![1306_Quadratic expression5.png](https://www.expertsmind.com/../CMSImages/1306_Quadratic expression5.png)
Case II: If a > 0
Sub case A: a < 0 and b2 - 4ac < 0 ⇔ f(x) < 0 ∀ x ∈ R.
In this type the parabola stays open downward and usually below the x-axis.
![232_Quadratic expression6.png](https://www.expertsmind.com/../CMSImages/232_Quadratic expression6.png)
Sub case B: a < 0 and b2 - 4ac = 0 ⇔ f(x) ≤ 0 ∀ x ∈ R.
In that type the parabola contact with the x - axis and stays open downward.
![102_Quadratic expression7.png](https://www.expertsmind.com/../CMSImages/102_Quadratic expression7.png)
Sub case C: a < 0 and b2 - 4ac > 0.Let f(x) = 0 have two real roots α and β (α < β).Thenf(x) <0 ∀ x ∈ (-∞, α)∪(β, ∞)and f(x) > 0 ∀ x ∈ (α, β).
![124_Quadratic expression8.png](https://www.expertsmind.com/../CMSImages/124_Quadratic expression8.png)
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 ![2249_Quadratic expression9.png](https://www.expertsmind.com/CMSImages/2249_Quadratic%20expression9.png)
Illustration: If min {x2 + (a - b)x + (1 - a - b)} > max (-x2 + (a +b)x - (1 + a + b)) Show that a2 + b2 < 4
Solution: Provided min{x2 + (a - b)x + (1 - a - b)} > max{-x2 + (a +b)x - (1 + a + b)}
![65_Quadratic expression10.png](https://www.expertsmind.com/CMSImages/65_Quadratic%20expression10.png)
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