JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Rational Algebraic Inequations

(1) Values of rational expression \[P(x)/Q(x)\] for real values of \[x,\] where \[P(x)\] and \[Q(x)\] are quadratic expressions : To find the values attained by rational expression of the form \[\frac{{{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}}{{{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}}\] for real values of \[x,\] the following algorithm will explain the procedure :

Algorithm

Step I: Equate the given rational expression to \[y\].

Step II: Obtain a quadratic equation in \[x\] by simplifying the expression in step I.

Step III: Obtain the discriminant of the quadratic equation in Step II.

Step IV: Put Discriminant \[\ge 0\] and solve the inequation for \[y\]. The values of \[y\] so obtained determines the set of values attained by the given rational expression.

(2) Solution of rational algebraic inequation: If \[P(x)\] and \[Q(x)\] are polynomial in \[x,\] then the inequation

\[\frac{P(x)}{Q(x)}>0,\,\frac{P(x)}{Q(x)}<0,\,\frac{P(x)}{Q(x)}\ge 0\] and \[\frac{P(x)}{Q(x)}\le 0\]

are known as rational algebraic inequations.

To solve these inequations we use the sign method as explained in the following algorithm.

Algorithm

Step I: Obtain \[P(x)\] and \[Q(x)\].

Step II: Factorize \[P(x)\] and \[Q(x)\] into linear factors.

Step III: Make the coefficient of \[x\] positive in all factors.

Step IV: Obtain critical points by equating all factors to zero.

Step V: Plot the critical points on the number line. If there are \[n\] critical points, they divide the number line into \[(n+1)\] regions.

Step VI: In the right most region the expression \[\frac{P(x)}{Q(x)}\] bears positive sign and in other regions the expression bears positive and negative signs depending on the exponents of the factors.

(3) Lagrange’s identity

If \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,{{b}_{1}},\,{{b}_{2}},\,{{b}_{3}}\in R\] then 

\[(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})-{{({{a}_{1}}{{b}_{1}}+{{a}_{2}}{{b}_{2}}+{{a}_{3}}{{b}_{3}})}^{2}}\]

\[={{({{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}})}^{2}}+{{({{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}})}^{2}}+{{({{a}_{3}}{{b}_{1}}-{{a}_{1}}{{b}_{3}})}^{2}}\]