Imaginary numbers.How are used imaginary numbers in solving quadratic equations?
Imaginary numbers are involved in solving of the quadratic equations, when the discriminant of the equation is negative.
The quadratic equation:
ax^2 + bx + c = 0
x1 = [-b+sqrt(delta)]/2a
x2 = [-b-sqrt(delta)]/2a
Discriminant of the quadratic equation:
delta = b^2 - 4ac
If delta is negative, then sqrt -delta = i*sqrt delta.
We have used the property of imaginary unit i:
i^2 = -1
i = sqrt(-1)
We'll solve an example:
x^2 - 4x + 8 = 0
delta = (-4)^2 - 4*1*8
delta = 16 - 32
delta = -16
sqrt delta = sqrt (-1)*(16) = i*sqrt 16 = 4i
The roots of the quadratic equation are:
x1 = (4+4i)/2
x1 = 2 + 2i
x2 = 2 - 2i
We remark here the fact that if an equation has a complex root, then the equation has as root the conjugate of the complex root, also.