The quadratic equations are of the form ax2+bx+c =0,
where x represents an unknown, and a, b, and c are constants with a not equal to 0.
Discriminant of a Quadratic Equation: The discriminant of a quadratic equation is defined as the number
D < 0 two roots which are complex conjugates
D = 0 one real root of multiplicity two
D > 0 two distinct real roots
D = positive perfect square two distinct rational roots (assumes a, b and c are rational)
- The quantity D(D=b2 – 4ac) is known as the discriminant of the quadratic equation.
- The quadratic equation has real and equal roots if and only if D = 0 i.e. b2 – 4ac = 0.
- The quadratic equation has real and distinct roots if and only if D > 0 i.e. b2 – 4ac . 0.
- The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 i.e. b2 – 4ac < 0.
- If p + iq (p and q being real) is a root of the quadratic equation where i = , then p – iq is also a root of the quadratic equation.
- If p + is an irrational root of the quadratic equation, then p – is also root of the quadratic equation provided that all the coefficients are rational.
- The quadratic equation has rational roots if D is a perfect square and a, b, c are rational. If a = 1 and b, c are integers and the roots of the quadratic equation are rational, then the roots must be integers.
- If the quadratic equation is satisfied by more than two numbers (real or complex), then it becomes an identity i.e. a = b = c = 0.
- Let a and b be two roots of the given quadratic equation. Then a + b = –b/a and ab = c/a.
- A quadratic equation, whose roots are a and b can be written as (x – a)(x – b) = 0 i.e., ax2 + bx + c º a(x – a)(x – b).
Depending on the sign of a and b2 –4ac, f(x) may be positive negative or zero. This gives rise to the following cases:
Basic Intervals in which Roots lie
In some problems we want the roots of the equation ax2 + bx + c = 0 to lie in a given interval. For this we impose conditions on a, b, and c. Let f(x) = ax2 + bx + c.
(i) If both the roots are positive i.e. they lie in (0, ¥), then the sum of the roots as well as the product of the roots must be positive.
Þa + b = – > 0 and ab = > 0 with b2 – 4ac > 0.
Similarly, if both the roots are negative i.e. they lie in (–¥, 0) then the sum of the roots will be negative and the product of the roots must be positive.
i.e. a + b = < 0 and ab = > 0 with b2 – 4ac > 0.
(ii) Both the roots are greater than a given number k if the following three conditions are satisfied D > 0, > k and a.f(k) > 0.
(iii) Both the roots will be less than a given number k if the following conditions are satisfied: D > 0, < k and a.f(k) > 0.
(iv) Both the roots will lie in the given interval (k1, k2) if the following conditions are satisfied D > 0, k1 < < k2 and a.f(k1) > 0,
a.f(k2) > 0.
(v) Exactly one of the roots lies in the given interval (k1, k2) if f(k1). f(k2) < 0.
(vi) A given number k will lie between the roots if a.f.(k) < 0.
In particular, the roots of the equation will be of opposite signs if 0 lies between the roots Þ a.f(0) < 0.