Step by step
The question
Question: which form of y = x² − 2x − 15 shows where it crosses the x axis?
1 Pick the form
A parabola can be written in three forms.
Each one shows one thing at a glance.
On the x axis, y is 0.
The factored form is a product, and a product is 0 when one bracket is 0.
So the factored form shows where it crosses the x axis.
2 Write it in factored form
Factoring goes backwards.
We need two numbers.
They multiply to −15, and they add to −2.
They give the brackets.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 15, smallest first.
We take the smaller from the bigger.
We need 2.
3 and 5 give 2.
With the signs: 3 and −5.
The two numbers go in the brackets: y = (x + 3)(x − 5).
That is the factored form.
3 Read the x axis crossings
On the x axis, y is 0.
So the product must be 0.
A product is 0 only when one of its parts is 0.
x + 3 is 0 when x = −3.
x − 5 is 0 when x = 5.
Careful!
Not 3.
x + 3 is 0 when x is −3: the sign in the bracket is the opposite.
So it crosses the x axis at x = −3 and x = 5: the points (−3, 0) and (5, 0).
4 Show it on the graph
Here is the graph of y = x² − 2x − 15.
Each square up is 2.
It crosses the x axis at −3 and 5.
5 Check
Let's check in the question's own form, y = x² − 2x − 15.
x = −3 gives y = 0: on the x axis.
And x = 5 gives y = 0: on the x axis.
So it is right!
Let's look back.
First: pick the form.
Then: write it in factored form.
Then: read the x axis crossings.
Then: show it on the graph.
Last: check.