SOLUTION: If 4 + I and 4 - I are roots of the equation z^2 + az + b = 0 find the value of a and the value of b?

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: If 4 + I and 4 - I are roots of the equation z^2 + az + b = 0 find the value of a and the value of b?      Log On


   

Question 1017147: If 4 + I and 4 - I are roots of the equation z^2 + az + b = 0 find the value of a and the value of b?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
If 4 + I and 4 - I are roots of the equation z^2 + az + b = 0 find the value of a and the value of b?
-------------------
(z - (4+i))*(z - (4-i)) = 0
z^2 - 8z + 17 = 0
a = -8, b = 17
=================
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-8x%2B17+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-8%29%5E2-4%2A1%2A17=-4.

The discriminant -4 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -4 is + or - sqrt%28+4%29+=+2.

The solution is x%5B12%5D+=+%28--8%2B-i%2Asqrt%28+-4+%29%29%2F2%5C1+=++%28--8%2B-i%2A2%29%2F2%5C1+, or
Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B17+%29