The roots of the quadratic equation

z^2+(4+i+qi)z+20=0 are w and w*
I) in the case where q is real, explain why q must be -1

clearly the roots of the given equation must be complex numbers, thus they must be conjugates of each other

let w = a+bi
then w* is a - bi

sum of roots = a+bi + a-bi = 2a
product or roots = (a+bi)(a-bi) = a^2 + b^2

but from the given equation:
sum of roots = -4-i-qi
product of roots = 20

a^2 + b^2 = 20, and
2a = -4-i-qi = -4 - (q+1)i
since a is the real part of w, 2a must be real
so in -4 - (q+1)i, the imaginary part must drop out, thus
q+1 = 0
q = -1

let's work backwards, suppose q = -1
then our equation simply becomes
z^2 + 4z+ 20 = 0
z^2 + 4z + 4 = -16
(z+2)^2 = -16
z+2 = ±4i
z = -2 ± 4i

so the sum of the roots = -2+4i + (-2 - 4i) = -4
product of roots = (-2+4i)(-2-4i)
= 4 - 16i^2 = 20

all is good!

Thank you so much

To find the roots of a quadratic equation, we can use the quadratic formula:

z = (-b ± √(b^2 - 4ac)) / (2a)

In this case, the equation is given as z^2 + (4 + i + qi)z + 20 = 0. Comparing this with the general form of the quadratic equation, we can determine the values of a, b, and c:

a = 1
b = (4 + i + qi)
c = 20

Now, we are given that the roots of the equation are w and w*. This means that when we substitute w into the equation, it should satisfy the equation and make it equal to zero. Similarly, if we substitute w* (the conjugate of w) into the equation, it should also satisfy the equation and make it equal to zero.

Substituting w into the equation, we get:
w^2 + (4 + i + qi)w + 20 = 0

Substituting w* into the equation, we get:
(w*)^2 + (4 + i + qi)w* + 20 = 0

Now, let's simplify these equations separately:
For w:
w^2 + (4 + i + qi)w + 20 = 0

For w*:
w*^2 + (4 + i + qi)w* + 20 = 0

Since w* is the conjugate of w, the coefficients of their corresponding terms will be the same except for the imaginary parts. Therefore, from the given equations, we can notice that the only difference between them is the coefficient of the imaginary part:

For w:
Coefficient of the imaginary part = 1 * q = q

For w*:
Coefficient of the imaginary part = 1 * q = -q (since w* is the conjugate of w)

Now, since the roots w and w* are derived from the same equation, they should satisfy both equations. This means that substituting w for w* and vice versa should still give the equation equal to zero:

Substituting w into the equation for w*, we get:
w^2 + (4 + i - qi)w + 20 = 0

Substituting w* into the equation for w, we get:
(w*)^2 + (4 + i - qi)w* + 20 = 0

Comparing these equations, we can observe that their imaginary parts have opposite signs. For w, the imaginary part is +qi, while for w*, the imaginary part is -qi.

Since w and w* are the roots of the same equation, but their imaginary parts have opposite signs, this implies that q must be -1.

So, in conclusion, when q is real, q must be equal to -1 for the roots of the quadratic equation z^2 + (4 + i + qi)z + 20 = 0 to be w and w*.