Step by step
The question
Question: write a quadratic with zeros −1 and −6.
zeros : −1, −6
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 −1 gives x − (−1).
Minus a minus is plus: x + 1.
x = −1 → x − (−1) = x + 1
Next, −6 gives x − (−6).
Minus a minus is plus: x + 6.
x = −6 → x − (−6) = x + 6
Careful!
The sign flips.
The zero −1 gives x + 1, because that is 0 when x is −1.
Careful: not zero −1 → (x − 1) ✗
2 Write f(x)
So f(x) = (x + 1)(x + 6).
f(x) = (x + 1)(x + 6)
3 Multiply out
Each part of the first bracket times each part of the second.
each part × each part
(x + 1) times (x + 6).
Each part times each part.
= x2 + 6x + x + 6
Like terms together: x² + 7x + 6.
= x2 + 7x + 6
So f(x) = x² + 7x + 6.
f(x) = x2 + 7x + 6
4 Check
Let's check: put each zero in.
First x = −1.
We get 0.
f(−1) = (−1)2 + 7(−1) + 6 = 1 − 7 + 6 = 0
And x = −6.
We get 0.
f(−6) = (−6)2 + 7(−6) + 6 = 36 − 42 + 6 = 0
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.
Answer
f(x) = x2 + 7x + 6