Step by step
The question
Question: write a quadratic with zeros −3 and 1.
zeros : −3, 1
A zero of f is an x that makes f(x) equal 0.
zero: an x where f(x) = 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.
zero c → bracket (x − c)
The zero −3 gives x − (−3).
Minus a minus is plus: x + 3.
x = −3 → x − (−3) = x + 3
Next, 1 gives x − 1.
x = 1 → x − 1
Careful!
The sign flips.
The zero −3 gives x + 3, because that is 0 when x is −3.
Careful: not zero −3 → (x − 3) ✗
2 Write f(x)
So f(x) = (x + 3)(x − 1).
f(x) = (x + 3)(x − 1)
3 Multiply out
Each part of the first bracket times each part of the second.
each part × each part
(x + 3) times (x − 1).
Each part times each part.
= x2 − x + 3x − 3
Like terms together: x² + 2x − 3.
= x2 + 2x − 3
So f(x) = x² + 2x − 3.
f(x) = x2 + 2x − 3
4 Check
Let's check: put each zero in.
First x = −3.
We get 0.
f(−3) = (−3)2 + 2(−3) − 3 = 9 − 6 − 3 = 0
And x = 1.
We get 0.
f(1) = 12 + 2(1) − 3 = 1 + 2 − 3 = 0
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.
Answer
f(x) = x2 + 2x − 3