The function f(x) = ax² + bx + c is called the quadratic expression. If a and b are the two distinct roots of the quadratic expression (or) A quadratic equation is an equation where the highest power of x is x². The standard form of quadratic equation is ax² + bx + c = 0.

Now, considering every possibility of the roots of the equation.

Case I: Roots are imaginary

Let one root of the quadratic equation be imaginary i.e. a = p + iq then b = p – iq and a (x – a) (x – b) = a [x – (p + iq)] [x – (p – iq)]

= a [(x – p) – iq] [(x – p) + iq]

= a [(x – p)2 + q2]

= a [(+) ve quantity]

So, sign of the expression depends on the sign of ‘a’, i.e. if a is   positive, then expression is positive. If a is negative, then expression is negative.

Case II: Roots are real and equal.

Let one root be a, then another root is also a.

Now, a(x – a) (x – b)    = a(x – a)²

= a [(+) ve quantity]

Again, the sign of the expression depends upon the sign of a.

Quadratic Formula: If α, β are roots of quadratic equation, then $$\alpha =\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$$, $$\beta =\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$$

Discriminant and Nature of Roots

D or Δ = b² – 4ac

If Δ > 0 and perfect square, then roots are real, rational and distinct

If Δ > 0 and not perfect square, then roots are real, irrational and distinct

If Δ = 0, roots are real and equal

If Δ < 0, then roots are complex conjugate numbers

Sum and Product of Roots

When α and β are roots,

$$\alpha \,+\,\beta \,=\,\frac{-b}{a}$$

$$\alpha \beta \,=\,\frac{c}{a}$$

Formation of quadratic equation with given roots, α and β

x² – (α + β) x b + αβ = 0.