Step by step
The question
Question: write a quadratic with zeros 1 + i and 1 − i.
A zero of f is an x that makes f(x) equal 0.
1 Each zero gives a bracket
If c is a zero, then x minus c is a factor.
Put c in, and that bracket is 0, so all of f(x) is 0.
The pair 1 + i and 1 − i gives two brackets.
Group x − 1 together in each bracket.
Then they look like A minus B, times A plus B.
A minus B, times A plus B, is A squared minus B squared.
And i squared is −1.
So we get (x − 1) squared, minus i squared.
i squared is −1.
Minus a minus is plus: x² − 2x + 2.
No i is left!
2 Write f(x)
So f(x) = (x² − 2x + 2).
3 Multiply out
Each part of the first bracket times each part of the second.
So f(x) = x² − 2x + 2.
4 Check
The pair: put x = 1 + i into x² − 2x + 2.
Remember i² = −1.
The i parts cancel, and the numbers make 0.
It works!
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.