If c is a real number and if 1 + i is a solution of the equation x^2 - 2x + c = 0, what is the value of c?

c = 2.

Step-by-step explanation:

Complex solutions come in conjugate pairs so the other solution is 1 - i.

So we have:

(x - (1 + i) (x - (1 - i) = 0

(x - 1 - i) (x - 1 + i) = 0

x^2 - x + ix - x + 1 - i - ix + i - i^2 = 0

x^2 - 2x + 1 - i^2 = 0

x^2 - 2x + 1 + 1 = 0

x^2 - 2x + 2 = 0

c = 2.

c = 2

Step-by-step explanation:

Substitute x = 1 + i into the equation

(1 + i) ² - 2 (1 + i) + c = 0 ← distribute left side

1 + 2i + i² - 2 - 2i + c = 0 (note that i² = - 1)

1 + 2i - 1 - 2 - 2i + c = 0 ← collect like terms on left

- 2 + c = 0 (add 2 to both sides)

c = 2