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You know that √(−9) = 3i. Then a JEE question asks you to solve a quadratic whose discriminant is −3 + 4i, and the quadratic formula suddenly needs the square root of a complex number. This post gives a four-line method, the sign rule that decides the answer, the polar shortcut and a full JEE-style example.
Where is this in the syllabus?
The NCERT Class 11 chapter Complex Numbers and Quadratic Equations (2026-27 reprint) covers square roots of negative real numbers in section 4.3.6, showing that the square roots of −3 are √3 i and −√3 i, and that the rule √a × √b = √(ab) needs checking when a and b are both negative. The general method for a + ib goes one step further.
The JEE (Main) 2026 syllabus lists "algebra of complex numbers" and "quadratic equations in real and complex number systems" in Unit 2. The JEE Advanced 2026 brochure adds "polar representation" and "properties of modulus and principal argument". Check the syllabus for your exam year.
How many square roots does a complex number have?
Exactly two, if the number is not zero, and they are negatives of each other. If w² = z, then (−w)² = z too. So the answer to "find the square roots of z" always has a ± in front.
What is the four-line method?
Let (x + iy)² = a + ib, with x and y real.
Expand: x² − y² + 2ixy = a + ib, so x² − y² = a and 2xy = b.
Use the identity (x² + y²)² = (x² − y²)² + (2xy)² = a² + b². So x² + y² = √(a² + b²), the modulus |z|.
Add and subtract: x² = (|z| + a)/2 and y² = (|z| − a)/2.
Fix the signs from 2xy = b. If b > 0, x and y have the same sign. If b < 0, they have opposite signs.
Worked example 1: a negative imaginary part
Find the square roots of −7 − 24i.
|z| = √(49 + 576) = √625 = 25.
x² = (25 − 7)/2 = 9, so x = ±3.
y² = (25 + 7)/2 = 16, so y = ±4.
b = −24 < 0, so the signs are opposite.
Answer: ±(3 − 4i). Check: (3 − 4i)² = 9 − 24i + 16i² = 9 − 16 − 24i = −7 − 24i.
Worked example 2: a positive imaginary part
Find the square roots of 5 + 12i. Here |z| = 13, so x² = 9 and y² = 4. Since b > 0, the signs match: ±(3 + 2i). Check: 9 − 4 + 12i = 5 + 12i.
When is the polar form quicker?
If z = r(cos θ + i sin θ), its square roots are ±√r (cos θ/2 + i sin θ/2). This is fastest when the argument is a standard angle.
For example, i has r = 1 and θ = π/2. Its square roots are ±(cos π/4 + i sin π/4) = ±(1 + i)/√2. Check: (1 + i)²/2 = (1 + 2i − 1)/2 = i.
Worked example 3: a quadratic with complex coefficients
Solve z² − 3z + (3 − i) = 0.
Discriminant: D = 9 − 4(3 − i) = 9 − 12 + 4i = −3 + 4i.
Square root of D: |D| = 5, x² = (5 − 3)/2 = 1, y² = (5 + 3)/2 = 4, and b > 0, so √D = ±(1 + 2i).
Roots: z = [3 ± (1 + 2i)]/2, giving 2 + i and 1 − i.
Check with the relations between roots and coefficients: the sum is 3 and the product is (2 + i)(1 − i) = 2 − 2i + i − i² = 3 − i. Both match the equation. Note that the roots are not conjugates. That only has to happen when every coefficient is real.
Which mistakes cost marks?
Forgetting the square root in |z|. x² + y² equals √(a² + b²), not a² + b².
Skipping the sign rule and writing ±3 ± 4i as four answers.
Giving one root. "The square roots" means both, with ±.
Multiplying roots of negatives. √(−4) × √(−9) is 2i × 3i = −6, not √36 = 6, as the NCERT section warns.
Expecting conjugate roots when the quadratic has complex coefficients, as in example 3.
For more ways to read complex numbers as points and rotations, see our guide to complex numbers as a geometry tool, and for the next step in roots, cube roots of unity.
If modulus and argument still feel shaky, Fermi's Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revisit the polar form before your next practice set.
A practice routine
Find the square roots of five numbers with integer answers (try 3 + 4i, −5 + 12i, 8 − 6i, −15 − 8i, 2i), checking each by squaring.
Do two by the polar method and compare the time taken.
Solve three quadratics with complex discriminants and check each with sum and product of roots.
Write down every sign error you make and the value of b in that question.
Start now: find the square roots of −15 − 8i and square your answer to check it.
