JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Some Important Result

(1) For the quadratic equation \[a{{x}^{2}}+bx+c=0\].

(i) One root will be reciprocal of the other if \[a=c\].

(ii) One root is zero if \[c=0\].

(iii) Roots are equal in magnitude but opposite in sign if \[b=0\].

(iv) Both roots are zero if \[b=c=0\].

(v) Roots are positive if \[a\] and \[c\] are of the same sign and \[b\] is of the opposite sign.

(vi) Roots are of opposite sign if \[a\] and \[c\] are of opposite sign.

(vii) Roots are negative if \[a,b,c\] are of the same sign.

(2) Let \[f(x)=a{{x}^{2}}+bx+c\], where \[a>0\]. Then

(i) Conditions for both the roots of \[f(x)=0\] to be greater than a given number \[k\] are \[{{b}^{2}}-4ac\ge 0;\,f(k)=0;\,\frac{-b}{2a}>k\].

(ii) Conditions for both the roots of \[f(x)=0\] to be less than a given number \[k\] are \[{{b}^{2}}-4ac\ge 0\], \[f(k)>0,\] \[\frac{-b}{2a}<k\].

(iii) The number \[k\] lies between the roots of \[f(x)=0\], if \[{{b}^{2}}-4ac>0;f(k)<0\].

(iv) Conditions for exactly one root of \[f(x)=0\] to lie between \[{{k}_{1}}\] and \[{{k}_{2}}\] is \[f({{k}_{1}})f({{k}_{2}})<0,\,\,{{b}^{2}}-4ac>0\].

(v) Conditions for both the roots of \[f(x)=0\] are confined between \[{{k}_{1}}\] and \[{{k}_{2}}\] is \[f({{k}_{1}})>0,\,f({{k}_{2}})>0,{{b}^{2}}-4ac\ge 0\] and \[{{k}_{1}}<\frac{-b}{2a}<{{k}_{2}}\], where \[{{k}_{1}}<{{k}_{2}}\].

(vi) Conditions for both the numbers \[{{k}_{1}}\]and \[{{k}_{2}}\] lie between the roots of \[f(x)=0\] is \[{{b}^{2}}-4ac>0;\,f({{k}_{1}})<0;\,f({{k}_{2}})<0\].