Let a and b be real numbers. The complex number 4 - 5i is a root of the quadratic z^2 + (a + 8i) z + (-39 + bi) = 0. What is the other root?

I did a lot of work on hand and with wolfram alpha but it doesn't simplify itself easily at all....anyone have any help to offer? Thanks

never mind I figured it out with vieta's formula :)

To find the other root of the quadratic equation, we will use the fact that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term, and the product of the roots is equal to the constant term divided by the coefficient of the quadratic term.

Let's call the other root of the quadratic equation as z1.

We have the following information:

- The sum of the roots is equal to -(a + 8i).
- The product of the roots is equal to (-39 + bi).

Using the given information, we can set up a system of equations:

1) z1 + (4 - 5i) = -(a + 8i)
2) (4 - 5i) * z1 = (-39 + bi)

Let's solve this system step-by-step.

Step 1: Rearrange Equation 1 to solve for z1.
z1 = -(a + 8i) - (4 - 5i)
z1 = -a - 4i - 8i - 4 + 5i
z1 = -a - 7i - 4

Step 2: Substitute this value into Equation 2.
(4 - 5i) * (-a - 7i - 4) = (-39 + bi)

Step 3: Expand and simplify the equation.
-4a - 28i - 16 - 20i + 35i^2 + 20 = -39 + bi

Since i^2 = -1, we can simplify further:
-4a - 48i - 35 + 20 = -39 + bi

Step 4: Combine like terms.
-4a - 35 - 15i = -39 + bi

Step 5: Equate the real and imaginary parts of the equation to obtain:
-4a - 35 = -39
-15i = bi

From the first equation, we can solve for a:
-4a - 35 = -39
-4a = -4
a = 1

From the second equation, we can equate the imaginary parts:
-15 = b

Therefore, the other root of the quadratic equation is z1 = -a - 7i - 4 = -(1) - 7i - 4 = -5 - 7i.

To find the other root of the quadratic equation, we can use the fact that if a+bi is a root, then its conjugate, a-bi, is also a root. In this case, 4 - 5i is given as a root. Therefore, we can find the other root by finding the conjugate of 4 - 5i.

The conjugate of a complex number of the form a + bi is simply a - bi, where a and b are real numbers. So, the conjugate of 4 - 5i is 4 + 5i.

Therefore, the other root of the quadratic equation is 4 + 5i.