Complex Numbers: When the Quadratic Formula Hits a Negative Discriminant

The imaginary unit i = √(−1) extends the number line into a plane. Master arithmetic with complex numbers, the conjugate trick for division, and you will handle every "no real solution" quadratic on the CBE.

8 分钟 TEKS 4F Algebra 2

When real numbers run out, math invents new ones

The equation x² = −1 has no real solution — there's no real number that squared gives a negative. So mathematicians invented one: i, defined so that i² = −1. Once you accept i, every quadratic has exactly two roots, and the entire algebraic universe stays consistent.

Big idea

Complex numbers have the form a + bi: a real part and an imaginary part. Treat i like a variable, with the rule i² = −1. Everything else follows.

Practice

The defining equation

What is i², where i is the imaginary unit?

Open the question →

Powers of i — the cycle of four

i¹ = i i² = −1 i³ = i² · i = −i i⁴ = (i²)² = 1 (then it repeats) Divide the exponent by 4 and look at the remainder: 0 → 1, 1 → i, 2 → −1, 3 → −i.

Arithmetic with complex numbers

Add / subtract: Combine real with real, imaginary with imaginary. (3 + 4i) + (1 − 2i) = 4 + 2i.
Multiply: Distribute (FOIL), then replace i² with −1. (1 + i)(2 + 3i) = 2 + 3i + 2i + 3i² = 2 + 5i − 3 = −1 + 5i.
Conjugate: Flip the sign of the imaginary part: conjugate of (a + bi) is (a − bi). Used to simplify division.

Division: multiply by the conjugate

To simplify a fraction with i in the denominator, multiply top and bottom by the conjugate of the denominator. The result is a real denominator.

(1 + i) / (1 − i) = (1 + i)(1 + i) / [(1 − i)(1 + i)] (multiply by conjugate) = (1 + 2i + i²) / (1 − i²) = (1 + 2i − 1) / (1 + 1) = 2i / 2 = i

When the discriminant is negative

From the previous lesson: discriminant < 0 means no real roots. With complex numbers, those roots exist — they're just imaginary.

x² + 4x + 5 = 0 (discriminant = 16 − 20 = −4) x = (−4 ± √−4) / 2 √−4 = √4 · √−1 = 2i x = (−4 ± 2i) / 2 = −2 ± i Complex roots always come in conjugate pairs (a + bi and a − bi) for polynomials with real coefficients.

3-second recap

i = √(−1), so i² = −1. The defining equation.
Add/subtract: combine matching parts. Multiply: FOIL, then replace i².
Divide: multiply by the conjugate of the denominator.
Negative discriminant → complex roots in conjugate pairs.