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.
- سوريا: الصف ١٢
- أفغانستان: الصف ١٠
- روسيا: الصف ١١
- فرنسا: الصف ١٢
- بريطانيا: السنة ١٢
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
A quadratic ax٢ + bx + c = ٠ has real roots only when the discriminant b٢ − ٤ac 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: √(−١٦) = ٤i.
The roots are x = −b±√(b٢ − ٤ac)٢a. When the discriminant is negative, the two roots have the same real part −b٢a 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.