Real roots of the equation x²(2p–1)x p= 0

Introduction:
The given equation is of quadratic form. The roots of the equation will be real if the discriminant of the quadratic equation is greater than or equal to zero.

Discriminant of the Quadratic Equation:
The discriminant of a quadratic equation ax² + bx + c = 0 is given by the expression b² – 4ac. If the discriminant is greater than zero, then the quadratic equation has two distinct real roots. If the discriminant is equal to zero, then the quadratic equation has one real root. If the discriminant is less than zero, then the quadratic equation has no real roots.

Substitution:
Substituting the given values of the quadratic equation, we get:
p = 0, x = 0

Discriminant:
The discriminant of the given quadratic equation is given by the expression (2p – 1)² – 4(0)(p) = (2p – 1)²

Conclusion:
From the above expression, we can see that the discriminant is always greater than or equal to zero. Therefore, the given quadratic equation x²(2p–1)x p= 0 will always have real roots.

Hence, we can conclude that the roots of the equation x²(2p–1)x p= 0 are real for all values of p.