Step by step
The question
Question: write a quadratic with zeros 2 and 5 through (0, 20).
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 zero 2 gives x − 2.
Next, 5 gives x − 5.
Careful!
The sign flips.
The zero 2 gives x − 2, because that is 0 when x is 2.
2 Write f(x)
The zeros fix the brackets, but f can be stretched: a times the brackets.
The point tells us a.
3 Find a
Put x = 0 and f(x) = 20 in.
Work out each bracket: −2 and −5.
−2 times −5 is 10.
So 20 = 10a.
To remove times 10, we divide both sides by 10.
10 divided by 10 is 1.
So just a is left.
So a = 2.
We put a in front of the brackets.
4 Multiply out
Each part of the first bracket times each part of the second.
(x − 2) times (x − 5).
Each part times each part.
Like terms together: x² − 7x + 10.
2 times each part.
So f(x) = 2x² − 14x + 20.
5 Check
Let's check: put each zero in.
First x = 2.
We get 0.
And x = 5.
We get 0.
And the point: put x = 0 in.
We get 20, as it should be.
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: find a.
Then: multiply out.
Last: check.