When we say that the zeros of a quadratic polynomial are equal, it means that the quadratic polynomial has only one root. This can happen in the following two cases:



The discriminant of a quadratic polynomial is given by the expression b² - 4ac. If the discriminant is equal to zero, then the roots of the quadratic polynomial are equal.



If the quadratic polynomial has only one root and it is not a real number, then the root must be a complex number of the form a + bi, where a and b are real numbers and i is the imaginary unit.

In this case, the quadratic polynomial can be written as (x - (a + bi))(x - (a - bi)) = (x - a - bi)(x - a + bi) = x² - 2ax + (a² + b²).

Comparing this with the general form of a quadratic polynomial ax² + bx + c, we get a = 1, b = -2a and c = a² + b².

Substituting the value of b in terms of a, we get c = a² + (-2a)² = 5a².

Therefore, the quadratic polynomial with equal roots can be written as x² - 2ax + 5a².

If the zeroes of the quadratic polynomial ax2+bx+c, where c≠0, are equal, then. The zeroes of the given quadratic polynomial ax2+bx+c,c≠0 are equal. If coefficient of x2 and constant term have the same sign i.e., c and a have the same sign.