The roots of the quadratic equation z^2 + az + b = 0 are 2 - 3i and 2 + 3i. What is a+b
(z - 2 +3i)(z -2 -3i) = 0
commutative property to multiply
z (z -2 -3i) = z^2 -2 z -3 z i
-2(z -2 -3i) = -2 z + 4 + 6 i
3i(z -2 -3i) = +3zi -6 i + 9
add ------------------------------
= z^2 - 4 z + 13
so
a = -4 and b = 9
I input 5 but I said it was wrong.
I input 5 but it said it was wrong
duh
there was a typo and you missed it. Always check the answers you find.
To find the sum of the roots of a quadratic equation, you can use the fact that the sum of the roots is equal to the negation of the coefficient of the linear term (a).
Given the roots 2 - 3i and 2 + 3i, we can see that they are complex conjugates. This means that if one root is 2 - 3i, the other root will be its conjugate, which is 2 + 3i.
To find a, we need to know the sum of these roots. Since the roots are complex conjugates, their sum will always be a real number. Therefore, the sum of the roots is:
(2 - 3i) + (2 + 3i) = 4
Now we know that the sum of the roots is 4.
According to the formula, the sum of the roots is equal to the negation of the coefficient of the linear term (a). So we have:
4 = -a
Therefore, we can determine that a = -4.
To find b, we can use the fact that the product of the roots is equal to the constant term (b) divided by the coefficient of the quadratic term (1 in this case).
The product of the roots is:
(2 - 3i)(2 + 3i) = 4 - 9i^2 = 4 + 9 = 13
Now we have the product of the roots, which is 13.
According to the formula, the product of the roots is equal to b divided by 1. So we have:
13 = b
Therefore, b = 13.
To find a+b, we add the values of a and b:
a + b = -4 + 13 = 9
Hence, a + b equals 9.