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.

Note : – (1)For solutions of specific quadratic equation

Solution of Quadratic equation

(2) For relation between coefficients and zeros of polynomial

Polynomials

Summary
Article Name
Condition and Nature of roots
Description
If coefficients are real then nature of roots can be determined by Discriminant . If Discriminant is positive then roots are real and distinct , if Discriminant is zero than roots are real and equal and if Discriminant is negative then roots are complex. Complex roots exist in conjugate pairs.
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