Identity is a relation that is always true. A formula can be defined as an equation that describes a unique relationship between different variables. An equation and expression are differentiated using the equal sign. The one with the equal sign is an equation, and the other is an expression. The quadratic polynomial with degree 2 and equated to zero is a quadratic equation. The general form of the quadratic equation is ax2 + bx + c = 0, where a, b, c ∈ R and a not equal to 0. The roots or solution to a quadratic equation can be found using the quadratic formula x = [- b ± √b2 – 4ac] / 2a. The nature of the roots can be explained using the value of the discriminant. The discriminant is obtained by D = b2 – 4ac. The quadratic formula modifies as follows:
x = [- b ± √D] / 2a, where D = b2 – 4ac
3 Different Cases of Roots of a Quadratic Equation
1) When D > 0: Real and distinct
2) When D = 0: Real and equal
3) When D < 0: Imaginary and in pairs
4) When D is a perfect square, then the roots are rational.
5) When D is not a perfect square, then the roots are irrational.
The Sum and Product of Roots
Assume α and β to be the roots of a quadratic equation.
Sum of roots
α + β = (- b + √D) / 2a + (-b – √D) / 2a
= (- b / 2a ) + (√D / 2a) – (b / 2a) – (√D / 2a)
= – 2b / 2a
= – b / a
Product of roots
α * β = (- b + √D) / 2a * (- b – √D) / 2a
= { (- b)² – (√D)² } / (2a)²
= (b² – D) / 4a²
= (b² – b² + 4ac ) /4a² [D = b² – 4ac ]
= c / a
A quadratic equation can be formed using the sum and product of the roots.
x2 – [α + β] x + [α . β] = 0
How to Find the Location of Roots
Consider the quadratic equation ax2 + bx + c, where a > 0, then
Case 1: Both roots of the function f (x) = 0 are greater than any constant k0 are b2 – 4ac ≥ 0, f (k0) > 0, (- b / 2a) > k0
Case 2: Both roots of the function f (x) = 0 are less than any constant k0 are b2 – 4ac ≥ 0, f (k0) > 0, (- b / 2a) < k0
Case 3: The constant k0 lies between the roots of the function f (x) = 0, if f (k0) < 0.
Case 4: If exactly one of the roots of the function f (x) = 0 lies between k1 and k2, then f (k1) f (k2) < 0, b2 – 4ac > 0.
Case 5: If both the roots of the function f (x) = 0 are lying between k1 and k2, then f (k1) > 0, f (k2) > 0, b2 – 4ac ≥ 0 and k1 < (- b / 2a) < k2 where k1 < k2
Case 6: If both k1 and k2 lie between the roots of the function f (x) = 0, then f (k1) < 0, f (k2) < 0.
Solved Example
If x is real, the expression y = (x + 2) / (2x2 + 3x + 6) takes all value in the interval
A) (1 / 13, 1 / 3)
B) [-1 / 13, 1 / 3]
C) (-1 / 3, 1 / 13)
D) None of the above
Solution: (B) [- 1 / 13, 1 / 3]
2yx2 + 3yx + 6y = x + 2
⇒ 2yx2 + (3y – 1) x + 6y – 2 = 0
∴ x is real
D ≥ 0
⇒ (3y – 1)2 – 8y (6y – 2) ≥ 0
⇒ (3y – 1) (13y + 1) ≤ 0
y ∈ [- 1 / 13, 1 / 3]
Location of Roots – Video Lesson
Location of Roots – Video Lesson
Frequently Asked Questions
What are the conditions that both the roots of the quadratic equation ax2 + bx + c = 0, (a>0) are greater than a constant k?
For the roots of the quadratic equation ax2 + bx + c = 0, (a>0) to be greater than a constant k, the discriminant, D ≥ 0, -b/2a > k and f(k) > 0.
Give the conditions that the roots of the quadratic equation ax2 + bx + c = 0, (a>0) lie between a constant k.
If f(k) < 0, then the roots of the quadratic equation ax2 + bx + c = 0, (a>0) lie between a constant k.
Give the conditions that both the roots of the quadratic equation ax2 + bx + c = 0, (a>0) lie between two constants k1 and k2 (k1< k2).
If f(k1) > 0, f(k2) > 0, b2 – 4ac ≥ 0 and k1 < (- b/2a) < k2 (where k1 < k2), then both the roots of the quadratic equation ax2 + bx + c = 0, (a>0) lies between k1 and k2.
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