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.