Ancient mathematicians did make commendable contribution to algebra. Especially, they could recognize that the quadratic equations have two roots and include negative as well as irrational roots. However, they were unable to solve all quadratics because of lack of concept of square roots of negative numbers. Indian mathematician Brahmagupta recognised negative roots. Arya Bhatta, Bhaskara-I, Mahaveeracharya, Euler, J.L. Lagrange, George Boole and John Von Neumann and several others contributed for the development of study of Quadratic equations in algebra.
        Now, let us meticulously examine a quadratic expression, quadratic equation.
¤ An equation of the form ax2+bx+c = 0 ------- (1)
        a ≠ 0, a, b, c  IR is called quadratic equation. A root of (1) is a complex number α such that aα²+bα+c = 0 and b²-4ac = Δ is known as discriminant of (1).

¤ Sum of the two roots is .
    product of roots is  .

¤ NATURE OF ROOTS:

¤ If α, β are roots of an quadratic equation, the equation is (x-α) (x-β) = 0.
¤ Let f(x) = ax²+bx+c
                  = a [x²+   x+ ]
                  

Sign of Quadratic expression.

f(x) > 0 if Δ < 0 and a > 0

f(x) ≥ 0 if Δ = 0 and a > 0

f(x) < 0 if Δ < 0 and a < 0

f(x) ≤ 0 if Δ = 0 and a < 0

¤ Sign and domain of a Quadratic expression
    Let f(x) = ax²+bx+c, a ≠ 0. a, b, c  IR
    Let α, β be roots of f(x) = 0.

¤ Repeated Root:
f(x) = a(x-α)², a ≠ 0, a, b, c  IR has α as a repeated root.
f(x) = ax²+bx+c, a ≠ 0, a, b, c  IR has α as a repeated root if f(α) = 0 and f'(α) = 0.  

ommon Root:
     The Quadratic equations ax²+bx+c = 0, a₁x²+b₁x+c₁ = 0 has a common root 'α', if (a₁c-ac₁)² = (bc₁-b₁c) (ab₁-a₁b).

      

Quadratic Expression: An expression of the  form ax² + bx + c where b and c are complex  numbers, is called a quadratic expression in x. Here a, b, c are called coefficients.
Ex: 5x² − 2x + 3, 2x² + 3x − 5
Quadratic Equation: An equation of the form ax² + bx + c = 0 is called a quadratic equation. Here a, b, c are called coefficients. The roots of Quadratic Equations: The roots of the quadratic equation
ax² + bx + c = 0 are

Note: 1) If a, b be the roots of ax² + bx + c = 0 then sum of the roots

2) The quadratic equation whose roots are a, b is x² − x (a + b) + ab = 0
3) If a, b be the roots ax² + bx + c = 0 of ax² + bx + c = a(x − a) ( x − b ) then
Definition: b² − 4ac is called the discriminant of the quadratic equation.
ax² + bx + c = 0  We write b² − 4ac = ∆
Nature of the roots of quadratic equations For a, b, c ∊ Q, 
i) If b² − 4ac > 0 and is a perfect square then roots are real, rational and unequal
ii) If b² − 4ac > 0 and is not a perfect square, then the roots are real irrational and unequal.
iii) If b² − 4ac = 0, then the roots are real, rational and equal. The two roots are equql to 
iv) If b² − 4ac < 0 then the roots are conjugate complex numbers.

 Transformed equations
Let f(x) = ax² + bx + c = 0 be a quadratic equation. If a and b are its roots, then
 
i) is an equation whose roots are   

.
 
ii) f(x − k) = 0 is an equation whose roots are 

iii) f(−x ) = 0 is an equation whose roots are

iv) = 0 is an equation whose roots are 

_{SOLVED PROBLEMS}
1. Find the roots of the equation x²− 7x + 12 = 0 ?

2. Find the roots of the equation 2x² + 3x + 2 = 0
Sol. The roots of quadratic equation ax² + bx + c = 0 are

3. Find the roots of the equation − x² + x + 2 = 0

              are −1 and 2.

4. Form the quadratic equation whose roots are 2 and 5?

5. Form the quadratic equation whose roots  are 

6. Form the quadratic equation whose roots are  − 3 + 5i and − 3 − 5i?