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Condition for roots of quadratic equation. to be real or complex

# Condition for roots of quadratic equation.

Condition for roots of quadratic equation $ax^2+bx+c=0 \,where\,\,,a\neq 0\,\,,\, a,b,c \in R$

Let Discriminant of quadratic $ax^2+bx+c=0$ is given by $D=b^2-4ac$

(1) Condition for roots to be complex :-

The roots of equation are complex if $D \lt 0$ .

Complex roots exist in conjugate pairs of the form p + iq , p – iq , where $i=\sqrt{-1}$.

(2) Condition for roots to be real and equal :-

The roots of quadratic equation are real and equal if D = 0 .

Roots are $\alpha , \beta = -(\frac{b}{2a})$

(3) Condition for roots to be real and distinct :-

The roots of equation are real and distinct $D \gt 0$ .

Roots are $\alpha , \beta = (\frac{-b \pm \sqrt{D}}{2a} )$

(4) Condition for roots to be rational :-

If a , b , c $\in$ Q and discriminant $D=m^2$ where m$\in$ Q then the roots are $\alpha , \beta =\frac{-b\pm m}{2a} \in Q$

since a , b , m$\in$ Q Where as if a , b , c $\in$ Q and discriminant $D\neq m^2$ where m$\in$Q then roots are irrational and exist in pairs of form p + $\sqrt{q}$ , p – $\sqrt{q}$.

(5) Condition for roots to be integers :-

If a = 1 , and b , c $\in$ I also discriminant $D=b^2-4c=m^2$ where m$\in$ I then roots $\alpha , \beta = (\frac{-b \pm m}{2}) \in I$

Since $b \pm m$ is an even integer as , 4c is always even hence if b is odd (or even ) integer m will be odd (or even ) integer respectively.