Question

Formula derivation for finding the roots of a quadratic equation

Answer 1

consider the quadratic equation
ax² + bx + c = 0 where a,b,c are real numbers
consider x₁ as on of the roots  of the quadratic equation
   ax₁² + bx₁ + c = 0
= a(x₁² + (b/a)x₁ + (c/a))
= a(x₁² + 2(b/2a)x₁ + b²/4a² + (c/a) - (b²/4a²))
= a((x₁+(b/2a))²+ (c/a) - (b²/4a²))
a((x₁+(b/2a))²+ (c/a) - (b²/4a²)) = 0
(x₁+(b/2a))²+ (c/a) - (b²/4a²) = 0
(x₁+(b/2a))² = (b²-4ac)/4a²
x₁+(b/2a) = (+/-)(b²-4ac)/2a
so, x₁ = -(b/2a)(+/-)(b²-4ac)/2a

if a, b, c are irrational numbers then 
the roots are usually irrational
and consider the expression  b²-4ac and if a, b, c are rational numbers and
i) b²-4ac>0 and a perfect square then the roots are rational numbers
ii) b²-4ac>0 and not a perfect square then the roots are irrational numbers
iii) b²-4ac=0 and the roots are same
iv) b²-4ac<0 then the roots are imaginary numbers