What is an example of a quadratic equation with imaginary roots?
1 Answer
If we consider a general quadratic equation:
# ax^2 + bx+ c = 0#
And suppose that we denote roots by
# x=alpha, beta => (x-alpha)(x-beta) = 0#
# :. x^2 - (alpha+beta)x+alpha beta = 0#
Equivalently we can write as
# :. x^2 - ("sum of roots")x+("product of roots") = 0#
And comparing these identical equations we can readily derive the following important relationships:
# "sum of roots" = -b/a# and# "product of roots" = c/a#
We also know that complex roots appear in conjugate pairs, so we can form some suitable equations.
Ex 1:
# S= (1-2i) + (1+2i) = 2 #
# P = (1-2i)(1+2i) = 1+4 = 5 #
So the equation with these roots is:
# x^2 - 2x+5 = 0#
Ex 2:
# S= (2-i) + (2+i) = 4 #
# P = (2-i)(2+i) = 4+1 = 5 #
So the equation with these roots is:
# x^2 - 4x+ 5= 0#
If we strictly answer the question and require imaginary roots then we have no real component so:
Ex 3:
# S= (-3i) + (3i) = 0 #
# P = (-3i)(3i) = 9 #
So the equation with these roots is:
# x^2 - 0x+ 9= 0#
# :. x^2 + 9= 0#
