The roots of #x^2 + 2x + c = 0# differ by #4i#. What are the roots, and what is the value of #c#?

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Answer 1 · 2016-01-28T04:33:20

#c=5# explanation is given below

This is an interesting question.

Let us think this out.

Let the two roots be #a+ib# and #a-ib#. Remember the complex roots come in conjugate pairs.

The sum of the roots is : #a+ib+a-ib = 2a#

The product of the roots is #(a+ib)(a-ib) = a^2+b^2#

Given the roots the quadratic equation can be formed using the following.

#x^2-"(Sum of the roots) "x + "(Product of the roots)" = 0#
Let us compare our given quadratic equation with this.
#x^2 +2x + c =0#

The sum of the roots #=-2#
Product of the roots #=c#

We know the sum of the roots when roots are #a+ib# and #a-ib# as #2a#

#2a = -2#
#a=-1#

Our question also informs us the difference of the roots is #4i#

#a+ib - (a-ib) = a+ib-a+ib = 2bi#

Given #2bi = 4i#
#2b=4#
#b=2#

The roots are #-1+2i# and #-1-2i#

The product of the roots #c=a^2+b^2 = (-1)^2+2^2 = 1+4=5#

#c=5#