JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Relations Between Roots and Coefficients

(1) Relation between roots and coefficients of quadratic equation : If \[\alpha \] and \[\beta \] are the roots of quadratic equation \[a{{x}^{2}}+bx+c=0\], \[(a\ne 0)\] then

Sum of roots \[=S=\alpha +\beta =\frac{-b}{a}=-\frac{\text{Coefficien}\text{t of }x}{\text{Coefficien}\text{t of }{{x}^{2}}}\]

Product of roots \[=P=\alpha .\beta =\frac{c}{a}=\frac{\text{Constant term}}{\text{Coefficient of }{{x}^{2}}}\]

(2) Formation of an equation with given roots : A quadratic equation whose roots are \[\alpha \] and \[\beta \] is given by \[(x-\alpha )(x-\beta )=0\].

\[\therefore \] \[{{x}^{2}}-(\alpha +\beta )x+\alpha \beta =0\]

i.e. \[{{x}^{2}}-(\text{sum of roots)}x+(\text{product of roots})=0\]

\[\therefore \] \[{{x}^{2}}-Sx+P=0\]

(3) Symmetric function of the roots : A function of \[\alpha \] and \[\beta \] is said to be a symmetric function, if it remains unchanged when \[\alpha \] and \[\beta \] are interchanged.

For example, \[{{\alpha }^{2}}+{{\beta }^{2}}+2\alpha \beta \] is a symmetric function of \[\alpha \] and \[\beta \] whereas \[{{\alpha }^{2}}-{{\beta }^{2}}+3\alpha \beta \] is not a symmetric function of  \[\alpha \] and \[\beta \].

In order to find the value of a symmetric function of \[\alpha \] and \[\beta \], express the given function in terms of \[\alpha +\beta \] and \[\alpha \beta \]. The following results may be useful.

(i)   \[a\]

(ii) \[{{\alpha }^{3}}+{{\beta }^{3}}={{(\alpha +\beta )}^{3}}-3\alpha \beta (\alpha +\beta )\]

(iii) \[{{\alpha }^{4}}+{{\beta }^{4}}=({{\alpha }^{3}}+{{\beta }^{3}})(\alpha +\beta )-\alpha \beta ({{\alpha }^{2}}+{{\beta }^{2}})\]

(iv) \[{{\alpha }^{5}}+{{\beta }^{5}}=({{\alpha }^{3}}+{{\beta }^{3}})({{\alpha }^{2}}+{{\beta }^{2}})-{{\alpha }^{2}}{{\beta }^{2}}(\alpha +\beta )\]

(v)  \[|\alpha -\beta |=\sqrt{{{(\alpha +\beta )}^{2}}-4\alpha \beta }\]

(vi) \[{{\alpha }^{2}}-{{\beta }^{2}}=(\alpha +\beta )(\alpha -\beta )\]

(vii)  \[{{\alpha }^{3}}-{{\beta }^{3}}=(\alpha -\beta )[{{(\alpha +\beta )}^{2}}-\alpha \beta ]\]

(viii) \[{{\alpha }^{4}}-{{\beta }^{4}}=(\alpha +\beta )(\alpha -\beta )({{\alpha }^{2}}+{{\beta }^{2}})\]