Imaginary numbers.How are used imaginary numbers in solving quadratic equations?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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