Second degree equation
Solving equation ax²+bx+c = 0
Δ = b²  4ac x = (b ± √Δ)/2a

If Δ>0 two solutions (or roots).
If Δ = 0, two equal solutions
If Δ<0, no real solutions. The two solutions are conjugate complex numbers
x = (b ± i√Δ)/2a
The proof is elementary :
ax²+bx+c = 0 can be written as a(x + b/2a)²  ab²/4a² + c = 0 that is :
(x + b/2a)² = (b²  4ac)/4a² et x +b/2a = ±√Δ/2a
This method is known as "completing the squares".
Sum and product of roots, symetric functions of roots
The sum of the two roots ist b/a, the product is c/a
(equating the expression
a(x  x_{1})(x  x_{2}) with
ax²+bx+c)
Every symetric function of the two roots (that is unchanged when exchanging the roots)
can be written as a function of the
sum S = b/a and the product
P = c/a.
For instance x
_{1}²+x
_{2}²=(x
_{1}+x
_{2})²2x
_{1}x
_{2}=S²2P
A little bit more difficult is x
_{1}x
_{2}
(
Solution)
or cos(x
_{1}x
_{2})
(
Solution)
Equations with rational coefficients
If a, b and c are all rational, multiplying by the common divisor, it is equivallent to a,b,c integers.
Then the solution will be rational if and only if
Δ = b²  4ac is a perfect square.
Else the roots are said to be "quadratic numbers"