Question

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?

Answer 1

c = 2.

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.

Answer 2

c = 2

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