If the highest degree coefficient is equal to 1 , then what is the total number of quadratic equations which are unchanged on squaring their roots ?

Step-by-step solution

Let the quadratic equation be x^2 + px + q = 0 . Let its roots be and . Sum of roots: + = -p Product of roots: = q The roots of the new equation are ^2 and ^2 . Sum of new roots: ^2 + ^2 = ( + )^2 - 2 = p^2 - 2q Product of new roots: ^2 ^2 = q^2 Since the equation remains unchanged, the sum and product of the roots must be the same as the original equation. Therefore, p^2 - 2q = -p and q^2 = q . From q^2 = q , we get q = 0 or q = 1 . Case 1: q = 0 Substituting q = 0 in p^2 - 2q = -p , we get p^2 = -p p^2 + p = 0 p(