Math Interactive
13. Complex Numbers
13.2. Complex algebra
13.2.2. Quadratic equations
Important

Quadratic equations

The usual quadratic formula still applies in complex numbers. In fact, we can now handle cases where the discriminant b2−4acb^2-4ac is negative as −1\sqrt{-1} can be now written as i.\mathrm{i}.

Examples

Step 1: Apply quadratic formula
z=−b±b2−4ac2a=2±22−4(2)(5)2(2)=2±−364 \begin{aligned} z &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ &= \frac{2 \pm \sqrt{2^2 - 4(2)(5)}}{2(2)} \\ &= \frac{2 \pm \sqrt{-36}}{4} \end{aligned}

Roots of quadratic equations

Working with real numbers, quadratic equations could have two real roots (including the case of a reapeated real root) or no real roots depending on the sign of the discriminant.

Extending our number system to the complex numbers mean we now have a cleaner results: all quadratic equations have two complex roots (including possible multiplicity).

(Note: recall that a real number can also be considered as a complex number.)

This result will be extended in section 3.1.4.

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