The roots of x^2 + 2x + c = 0x2+2x+c=0 differ by 4i4i. What are the roots, and what is the value of cc?

1 Answer
Jan 28, 2016

c=5c=5 explanation is given below

Explanation:

This is an interesting question.

Let us think this out.

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

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

The product of the roots is (a+ib)(a-ib) = a^2+b^2(a+ib)(aib)=a2+b2

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

x^2-"(Sum of the roots) "x + "(Product of the roots)" = 0x2(Sum of the roots) x+(Product of the roots)=0
Let us compare our given quadratic equation with this.
x^2 +2x + c =0x2+2x+c=0

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

We know the sum of the roots when roots are a+iba+ib and a-ibaib as 2a2a

2a = -22a=2
a=-1a=1

Our question also informs us the difference of the roots is 4i4i

a+ib - (a-ib) = a+ib-a+ib = 2bia+ib(aib)=a+iba+ib=2bi

Given 2bi = 4i2bi=4i
2b=42b=4
b=2b=2

The roots are -1+2i1+2i and -1-2i12i

The product of the roots c=a^2+b^2 = (-1)^2+2^2 = 1+4=5c=a2+b2=(1)2+22=1+4=5

c=5c=5