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 $c=a^2+b^2 = (-1)^2+2^2 = 1+4=5$

$c=5$

The roots of x^2 + 2x + c = 0x^2 + 2x + c = 0 differ by 4i4i. What are the roots, and what is the value of cc?