JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Wavy Curve Method

Let \[f(x)={{(x-{{a}_{1}})}^{{{k}_{1}}}}{{(x-{{a}_{2}})}^{{{k}_{2}}}}{{(x-{{a}_{3}})}^{{{k}_{3}}}}......\]\[{{(x-{{a}_{n-1}})}^{{{k}_{n-1}}}}{{(x-{{a}_{n}})}^{{{k}_{n}}}}\]                                         …..(i)

where \[{{k}_{1}},\,{{k}_{2}},\,{{k}_{3}}...,\,{{k}_{n}}\in N\] and \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,......,\,{{a}_{n}}\] are fixed natural numbers satisfying the condition \[{{a}_{1}}<{{a}_{2}}<{{a}_{3}}.....<{{a}_{n-1}}<{{a}_{n}}\]

First we mark the numbers \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,......,\,{{a}_{n}}\] on the real axis and the plus sign in the interval of the right of the largest of these numbers, i.e. on the right of \[{{a}_{n}}\]. If \[{{k}_{n}}\] is even then we put plus sign on the left of \[{{a}_{n}}\] and if \[{{k}_{n}}\] is odd then we put minus sign on the left of \[{{a}_{n}}\]. In the next interval we put a sign according to the following rule :

When passing through the point \[{{a}_{n-1}}\] the polynomial \[f(x)\] changes sign if \[{{k}_{n-1}}\] is an odd number and the polynomial \[f(x)\] has same sign if \[{{k}_{n-1}}\] is an even number. Then, we consider the next interval and put a sign in it using the same rule. Thus, we consider all the intervals. The solution of \[f(x)>0\] is the union of all intervals in which we have put the plus sign and the solution of \[f(x)<0\] is the union of all intervals in which we have put the minus sign.