ACT® · Math: Number and Quantity
Equations with Complex Roots
A quadratic that never crosses the x-axis has a negative discriminant, so the quadratic formula still gives two answers — they just come out as a conjugate pair of complex numbers, p + qi and p - qi.
Every quadratic ax2 + bx + c = 0 has two solutions from the quadratic formula, x = (-b ± √(b2 - 4ac)) / (2a). The part under the radical, b2 - 4ac, is the discriminant, and it decides what kind of answers you get. When it's positive, you get two real, different x-intercepts. When it's zero, the graph touches the axis once. When it's negative, the graph never reaches the axis at all — but the formula doesn't stop working. It just means you're taking the square root of a negative number.
That's where i comes in: √(-1) = i. If the discriminant is negative, pull the negative sign out: √(4ac - b2) becomes i*√(4ac - b2), since 4ac - b2 is now positive. The ± in the formula then splits into two mirror-image answers, p + qi and p - qi, where p is -b/(2a) and q is √(4ac - b2)/(2a). These are called complex conjugates — same real part, opposite sign on the imaginary part — and they always arrive as a pair because of that ± sign.
This connects to the bigger idea that 'no real roots' isn't 'no roots.' The graph is a picture of real-number behavior only; it can't show solutions that live off the real line. Algebraically the equation still has exactly two roots, counted with the complex numbers included, matching the pattern that a degree-2 polynomial always has two solutions, real or not.
Key terms
- discriminant
- The expression b2 - 4ac inside the quadratic formula's square root, which tells you the type of roots.
- complex conjugate pair
- Two complex numbers of the form p + qi and p - qi that share a real part and have opposite imaginary parts.
- imaginary unit i
- The number defined by i2 = -1, used to write square roots of negative numbers.
- nonreal solution
- A root of an equation that is a complex number with a nonzero imaginary part, so it doesn't appear on the real x-axis.
Watch it explained
- How it worksHow does a negative number under the radical turn the quadratic formula's answers into a conjugate pair?
- Worked exampleFind both solutions of x2 + 4x + 13 = 0 in a + bi form.
- Worked exampleFind both solutions of x2 + 25 = 0 and show where the plus-or-minus comes from.
- Common mistakeWhy isn't a quadratic with no x-intercepts a quadratic with no roots whatsoever?
Check yourself
End of topic.