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Common Mistakes

Traps in Complex Numbers & Quadratic Equations

10 mistake patterns students fall for. 4 high-frequency traps appear in almost every exam.

Forgetting conjugate when dividing

Very CommonFORMULA

When dividing $z_1/z_2$ , students forget to multiply numerator and denominator by the conjugate of the denominator.

Why: Division looks simple but requires rationalizing the denominator to separate real and imaginary parts.

WRONG: Directly dividing:

\frac{a+bi}{c+di} = \frac{a}{c} + \frac{b}{d}i

RIGHT: Multiply by

\frac{\bar{z}_2}{\bar{z}_2}

\frac{(a+bi)(c-di)}{c^2+d^2}

Wrong quadrant for argument

Very CommonSIGN ERROR

Using $\tan^{-1}(b/a)$ directly gives principal value in $(-\pi/2, \pi/2)$ , which is wrong for $z$ in Q2 or Q3.

Why: The calculator gives the reference angle, but arg depends on which quadrant $z$ lies in.

WRONG:

\arg(-1+i) = \tan^{-1}(1/(-1)) = -\pi/4

RIGHT:

\arg(-1+i) = \pi - \pi/4 = 3\pi/4

(Q2, so add

\pi

to the reference angle)

Writing $z \cdot \bar{z} = z^2$ instead of $|z|^2$

Very CommonFORMULA

$z \cdot \bar{z} = |z|^2$ (a real number), not $z^2$ . This is one of the most common errors.

Why: Students confuse multiplying by conjugate with squaring because both involve two copies of $z$ .

WRONG:

z \cdot \bar{z} = z^2 \Rightarrow (3+4i)(3-4i) = (3+4i)^2 = -7+24i

RIGHT:

z \cdot \bar{z} = |z|^2 \Rightarrow (3+4i)(3-4i) = 9+16 = 25

Expanding instead of using properties

Very CommonTIME WASTE

Many problems are solved faster using properties ( $|z|^2 = z\bar{z}$ , arg rules, geometry) than brute-force $x+iy$ substitution.

Why: Substituting $z = x + iy$ feels safe but creates unnecessarily long algebra.

WRONG: For

|z-1| = |z+1|

, substituting

z = x+iy

and expanding both sides

RIGHT: Geometric approach: equidistant from

1

and

-1

means

\text{Re}(z) = 0

. Done in 5 seconds.

Applying De Moivre without polar conversion

CommonFORMULA

Students try to apply De Moivre's theorem directly to $(a+bi)^n$ without first converting to $r(\cos\theta + i\sin\theta)$ form.

Why: Binomial expansion of $(a+bi)^n$ is tempting but extremely error-prone for large $n$ .

WRONG:

(1+i)^8

via binomial expansion (messy and error-prone)

RIGHT:

(1+i) = \sqrt{2}\,e^{i\pi/4} \Rightarrow (1+i)^8 = (\sqrt{2})^8 \cdot e^{i \cdot 2\pi} = 16

Confusing $|z-a| = |z-b|$ with a circle

CommonCASE MISS

$|z-a| = |z-b|$ is the perpendicular bisector (a line), not a circle. Students often set up the wrong locus.

Why: $|z-a| = r$ is a circle, so students assume any modulus equation gives a circle.

WRONG: Treating

|z-(1+i)| = |z-(2-i)|

as a circle equation

RIGHT: This is a straight line: the perpendicular bisector of the segment joining

(1,1)

and

(2,-1)

Conjugate pair rule has conditions

CommonDOMAIN

Complex roots come in conjugate pairs only when the polynomial has REAL coefficients. Not true for complex coefficients.

Why: The conjugate root theorem is taught without emphasizing the real-coefficient requirement.

WRONG: If

2+3i

is a root of

z^2 + (1+i)z + c = 0

, assuming

2-3i

is also a root

RIGHT: Conjugate pair rule does not apply here because

(1+i)

is not a real coefficient. Use sum/product of roots instead.

Sign error in conjugate operations

CommonSIGN ERROR

Students mess up signs when computing conjugates of sums, products, or quotients.

Why: Conjugate distributes over all operations, but the sign change applies to every $i$ in the expression.

WRONG:

\overline{\left(\frac{3-2i}{1+i}\right)} = \frac{3+2i}{1+i}

(forgot to conjugate denominator)

RIGHT:

\overline{z_1/z_2} = \bar{z}_1/\bar{z}_2

, so

\overline{\left(\frac{3-2i}{1+i}\right)} = \frac{3+2i}{1-i}

Missing roots when solving $z^n = w$

CommonCASE MISS

$z^n = w$ has exactly $n$ roots. Students often find only one root and forget the others.

Why: Real-number intuition (one $n$ th root) carries over incorrectly to complex numbers.

WRONG:

z^3 = 8 \Rightarrow z = 2

(only finding the real root)

RIGHT:

z^3 = 8 \Rightarrow z = 2,\, 2\omega,\, 2\omega^2

where

\omega = e^{i2\pi/3}

. All three roots must be listed.

Missing cross term in $|z_1 + z_2|^2$

OccasionalFORMULA

When squaring $|z_1 + z_2| = k$ , students expand incorrectly or forget the cross term.

Why: The real identity $(a+b)^2 = a^2 + 2ab + b^2$ does not directly apply to modulus.

WRONG:

|z_1+z_2|^2 = |z_1|^2 + |z_2|^2

(missing cross term)

RIGHT:

|z_1+z_2|^2 = |z_1|^2 + |z_2|^2 + 2\text{Re}(z_1\bar{z}_2)

Test yourself

Can you spot these traps under time pressure?

Take a timed quiz on Complex Numbers & Quadratic Equations and see if you avoid the mistakes above.

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Traps in Complex Numbers & Quadratic Equations

Forgetting conjugate when dividing

Wrong quadrant for argument

Writing z⋅zˉ=z2z \cdot \bar{z} = z^2z⋅zˉ=z2 instead of ∣z∣2|z|^2∣z∣2

Expanding instead of using properties

Applying De Moivre without polar conversion

Confusing ∣z−a∣=∣z−b∣|z-a| = |z-b|∣z−a∣=∣z−b∣ with a circle

Conjugate pair rule has conditions

Sign error in conjugate operations

Missing roots when solving zn=wz^n = wzn=w

Missing cross term in ∣z1+z2∣2|z_1 + z_2|^2∣z1​+z2​∣2

Writing $z \cdot \bar{z} = z^2$ instead of $|z|^2$

Confusing $|z-a| = |z-b|$ with a circle

Missing roots when solving $z^n = w$

Missing cross term in $|z_1 + z_2|^2$