1.10: Quadratic Equations in the Complex Number System

A quadratic equation in the form ax²+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b²−4ac. In this expression, a is the coefficient of the quadratic term x², b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.

The square root of a negative number is rewritten using i. For instance, instead of writing −36, it is expressed as 36i=6i. This adjustment allows solutions to be represented as complex numbers, which have both real and imaginary components.

To see how this works in context, consider the quadratic equation x²+6x+13=0 . The coefficients here are a=1, b=6, and c=13. Before applying the quadratic formula, it helps to examine the discriminant:

Because the discriminant is negative, the square root of -16 is written as 4i. Substituting into the quadratic formula,

The solutions, −3+2i and −3-2i, form a pair of complex conjugates.

Substituting one of these back into the original equation helps confirm its validity. Using −3+2i, the left-hand side becomes:

Expanding and simplifying this expression results in 0, verifying that the value satisfies the equation.

This example highlights how complex numbers extend the number system to ensure all quadratic equations have solutions. The discriminant continues to guide the nature of the solutions: positive for distinct real solutions, zero for a repeated real solution, and negative for complex conjugate pairs.

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.