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Quadratic Complex Equations

Level 1 - Math I (Physics) topic page in Complex Equations.

Principle

A quadratic complex equation has a squared unknown. For real coefficients, complex numbers make every quadratic solvable, even when the discriminant is negative.

Quadratic equation
\[az^2+bz+c=0,\quad a,b,c\in\mathbb R,\quad a\ne0\]

The complex number system contains square roots of negative real numbers because \(i^2=-1\).

Notation

  • \(z\) is the unknown complex number.
  • \(a\), \(b\), and \(c\) are real coefficients.
  • \(\Delta=b^2-4ac\) is the discriminant.
  • \(i\) is the imaginary unit, with \(i^2=-1\).

Method

Start with:

Start
\[az^2+bz+c=0\]

Multiply by \(4a\):

Multiply by 4a
\[4a^2z^2+4abz+4ac=0\]

Move the constant term:

Move constant
\[4a^2z^2+4abz=-4ac\]

Add \(b^2\) to both sides:

Complete square
\[4a^2z^2+4abz+b^2=b^2-4ac\]

Factor the left-hand side:

Squared form
\[(2az+b)^2=b^2-4ac=\Delta\]

Then solve for \(z\). If \(\Delta\lt0\), write \(\Delta=-|\Delta|\), so \(\sqrt{\Delta}=i\sqrt{|\Delta|}\).

Rules

Positive discriminant
\[\Delta\gt0\quad\Longrightarrow\quad z=\frac{-b\pm\sqrt{\Delta}}{2a}\]
Zero discriminant
\[\Delta=0\quad\Longrightarrow\quad z=-\frac{b}{2a}\]
Negative discriminant
\[\Delta\lt0\quad\Longrightarrow\quad z=\frac{-b\pm i\sqrt{-\Delta}}{2a}\]

For real coefficients, non-real roots occur in conjugate pairs. If \(u+iv\) is a root, then \(u-iv\) is also a root.

Examples

Solve:

Example equation
\[z^2+2z+2=0\]

Here \(a=1\), \(b=2\), and \(c=2\). The discriminant is:

Example discriminant
\[\Delta=b^2-4ac=2^2-4(1)(2)=4-8=-4\]

Because \(\Delta\lt0\), use the negative-discriminant form:

Example formula
\[z=\frac{-2\pm i\sqrt{4}}{2}\]

Simplify:

Example answer
\[z=-1\pm i\]

Checks

  • Confirm that the equation is actually quadratic by checking \(a\ne0\).
  • Use \(i\sqrt{-\Delta}\), not \(\sqrt{\Delta}\), when \(\Delta\lt0\).
  • For real coefficients, check that non-real roots are conjugates.
  • Substitute one root back into the original equation when signs are uncertain.

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Linear Complex Equations

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Roots of Unity