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 - x1)(x - x2) 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 x1²+x2²=(x1+x2)²-2x1x2=S²-2P
A little bit more difficult is |x1-x2| (Solution) or cos(x1-x2) (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"

 

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