Quadratic equation important properties

Important properties of a quadratic equation

Important properties of a quadratic equation
  a\(x^{2}\)+bx+c=0
1) A root of a quadratic equation is number '\(\alpha\)' (real or complex) such that  a\(\alpha^{2}+b \alpha + c=0\)

2) It has two roots real or imeginary.
   \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
             [Quadratic formula]
    a\(x^{2}\)+bx+c=0
    \(\therefore\) a(x-\(\alpha\))(x-\(\beta\))=0
    So, x=\(\alpha\), or x=\(\beta\))
     the quadratic formula 
             \(x=\frac{-b\pm\sqrt{D}}{2a}\)
3)(i) If D>0 and D is a perfect square, then roots are real, distinct and rational number.
   (ii) If D>0 and D is not a perfect square, then roots are real, distinct but irrational numbers. 
         \(x=a\pm\sqrt{b}\)
     (iii) If D=0, then roots are real and equal.
          \(x=\alpha=\beta=\frac{-b}{2a}\)
      (iv) If D<0 , they are distinct complex number with non zero imaginary part. 
         \(x=\alpha, \beta\) 
           x=\(p\pm iq\)

4) If p+iq (whee p and q are real numbs) is a root of a\(x^{2}\)+bx+c=0, then other root must be p-iq (conjugate of p+iq)

5) If \(p+\sqrt{q}\) (whee p and q are real numbs) is a root of a\(x^{2}\)+bx+c=0, then other         root must be \(p-\sqrt{q}\).
     Provided all the coefficients of x, i.e, a, b, and c are rational numbers.

6) If a\(x^{2}\)+bx+c=0 is satisfied by more than three numbers (real o complex), then it   becomes an identity.
      i.e, a=b=c=0

7) If \(\alpha\) and \(\beta\) are two roots of a quadratic equation, we will get the equation,         by using th relation
     \(x^{2}\)-(\(\alpha\)+\(\beta\))x+c=0
      i.e, \(x^{2}\)-(sum of zeos)x+(product of zeos)=0