Theory Of Equations

Given equation: Let Substituting, This is a reciprocal equation of the form: Since , divide throughout by : Rearranging, Let Substitute: Since , by AM–GM: Hence is rejected. So, Multiplying by : Both roots are positive, hence both are valid. Since , each value of gives one real value of .

Step 1: For a non-trivial solution, the determinant of the coefficients must be zero.
Step 2:

Step 3: This is a circulant determinant: .
Step 4: From : and .
Step 5: Case 1: (but ).
Step 6: Case 2: . .
Step 7: .

Step 1: Substitute .
Step 2: Condition for real and distinct roots.
Discriminant must be positive: Step 3: Roots of must be positive.
Product of roots and sum , so both roots are positive.
Step 4: Smallest integer satisfying the condition.

We want the least integer value of such that

has four real and distinct roots.

Put (so ). Then the equation becomes

For to have four real and distinct roots, this quadratic in must have two distinct positive roots (so that ).

• Distinct roots in : Discriminant .
• Positive roots in : Product and sum . For both to be positive, we need .

Combining: . Hence the least integer value is

.