Explanation
Let the quadratic equation be x2+px+q=0, where the coefficient of x2 is 1. Let its roots be α and β.
According to the properties of roots:
α+β=−p
αβ=q
If the equation remains unchanged when we square the roots, the new roots must be α2 and β2, which must be the same as the original roots α and β. There are two possibilities:
Case 1: The roots remain the same individually
This happens if α2=α and β2=β.
Possible sets for {α,β}:
{0,0}⟹ Equation: x2=0 (Here p=0,q=0)
{1,1}⟹ Equation: (x−1)2=0⟹x2−2x+1=0 (Here p=−2,q=1)
{0,1}⟹ Equation: x(x−1)=0⟹x2−x=0 (Here p=−1,q=0)
Case 2: The roots are swapped
This happens if α2=β and β2=α.
Substituting β=α2 into the second equation:
(α2)2=α⟹α4−α=0⟹α(α3−1)=0
α(α−1)(α2+α+1)=0
If α=0, then β=0 (Already covered in Case 1).
If α=1, then β=1 (Already covered in Case 1).
If α2+α+1=0, then the roots are the complex cube roots of unity, ω and ω2.
Summarizing the distinct quadratic equations found:
x2=0
x2−2x+1=0
x2−x=0
x2+x+1=0
There are 4 such quadratic equations.
Conclusion: The correct option is (b) 4.
