Question

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

Answer 1

`c=5` explanation is given below

Explanation:

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`