JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Properties of Quadratic Equation

(1) If \[f(a)\] and \[f(b)\] are of opposite signs then at least one or in general odd number of roots of the equation \[f(x)=0\] lie between \[a\] and \[b\].

(2) If \[f(a)=f(b)\] then there exists a point \[c\] between \[a\] and \[b\] such that \[{f}'(c)=0\], \[a<c<b\].

(3) If \[\alpha \] is a root of the equation \[f(x)=0\] then the polynomial \[f(x)\] is exactly divisible by \[(x-\alpha )\], then \[(x-\alpha )\] is factor of \[f(x)\].

(4) If the roots of the quadratic equations \[{{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\] and \[{{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}=0\] are in the same ratio \[\left( i.e.\,\,\frac{{{\alpha }_{1}}}{{{\beta }_{1}}}=\frac{{{\alpha }_{2}}}{{{\beta }_{2}}} \right)\]  then \[b_{1}^{2}/b_{2}^{2}={{a}_{1}}{{c}_{1}}/{{a}_{2}}{{c}_{2}}\].