JEE Maths Notes: Quadratic Equations

The quadratic equations are of the form  ax2+bx+c =0,
where x represents an unknown, and ab, 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= b2-4ac

Discriminant                                         Roots

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)

Basic Results

  1. The quantity D(D=b2 – 4ac) is known as the discriminant of the quadratic equation.
  2. The quadratic equation has real and equal roots if and only if D = 0 i.e. b2 – 4ac = 0.
  3. The quadratic equation has real and distinct roots if and only if D > 0 i.e. b2 – 4ac . 0.
  4. The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 i.e. b2 – 4ac < 0.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. Let a and b be two roots of the given quadratic equation. Then a + b = –b/a and ab = c/a.
  10. 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).

Pictorial Representations: 

Depending on the sign of a and b2 –4ac, f(x) may be positive negative or zero. This gives rise to the following cases:

(i) a > 0 and b2 –4ac < 0

Û f(x) > 0  x Î R.

In this case the parabola always

remains above the x-axis.

(ii) a > 0 and b2 –4ac = 0

Û f(x) > 0  x Î R.

In this case the parabola always

remains above the x-axis.

(iii) a > 0 and b2 – 4ac > 0.

Let f(x) = 0 have two real roots a and b (a<b)

Then f(x) > 0  x Î (–¥, a)È(b, ¥)

and f(x) < 0  x Î (a, b).

(iv) a < 0 and b2 – 4ac < 0

Û f(x) < 0  x Î R

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

(v) a < 0 and b2 – 4ac = 0

Û f(x) < 0  x Î R.

In this case the parabola touches the

x-axis and lies below the x-axis.

(vi) a < 0 and b2 – 4ac > 0

Let f(x)=0 have two real roots a and b (a<b).

Then f(x) < 0 x Î(–¥, a)È(b, ¥) and f(x) > 0  x Î (a, b).

 

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.

 

 

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