Solving incomplete quadratic equations

If the quadratic equation has the form ax² = 0, then it has a single root x = 0.

If the quadratic equation has the form ax² + c = 0, then ax² = -c and, therefore, x² = -c/a. The right side of the last equation is a number that, by the condition, is not equal to zero. Therefore, if the right side is less than zero, then the equation x² = -c/a, and, therefore, the original ax² + c = 0 has no roots. If the right side is greater than zero, then the equation x² = -c/a has two real roots x₁ = -√(-c/a), x₂ = √(-c/a).

If the quadratic equation is of the form ax² + bx = 0, then using the factorization method, we obtain ax² + bx = 0 => x(ax+b)=0 => x=0, ax+b=0 => x=0, x=-b/a. Thus, the original equation has two roots x₁=0 и x₂ = -b/a.