Kwadratische vergelijkingen in het complexe getallenstelsel

Quadratic equations in which the discriminant —the expression under the square root in the quadratic formula—has a negative value have no real solutions.

To solve these, the number system is extended to include complex numbers using the imaginary unit i, defined as the square root of a negative one.

The discriminant determines the solution's nature; a positive discriminant gives two unequal real solutions.

A zero discriminant gives one repeated real solution, meaning both solutions from the quadratic formula are equal.

A negative discriminant gives two complex solutions: non-real and complex conjugates—meaning, complex number pairs with equal real parts and imaginary parts with opposite sign.

An example of such a quadratic equation has solutions including the imaginary unit i, resulting from a negative discriminant.

Substituting these complex solutions into the original equation gives same values on both sides, confirming their validity.

Consider a rectangle with a perimeter of 20 cm, area of 30 cm². Substituting the perimeter equation into the area expression gives a quadratic equation with a negative discriminant, leading to complex roots—showing no real rectangle can exist.