Quadratics and polynomials with complex roots
See why x² + 4x + 13 = 0 has no real roots, solve equations whose roots are complex, plot them on the plane and find the missing root of a cubic.
- UK: Year 12
- France: grade 12
- Afghanistan: grade 10
- Syria: grade 12
- Russia: grade 11
Guide
A quadratic ax2 + bx + c = 0 has real roots only when the discriminant b2 − 4ac is zero or positive. If it is negative, the square root of a negative number is i times a real number, so the roots are complex: √(−16) = 4i.
The roots are x = −b±√(b2 − 4ac)2a. When the discriminant is negative, the two roots have the same real part −b2a and opposite imaginary parts: they are conjugates, mirror images in the real axis.
The same is true for any polynomial with real coefficients: complex roots come in conjugate pairs.