Step by step
The question
Question: which form of y = x² − 6x + 5 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 5, and they add to −6.
They give the brackets.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 5, smallest first.
We add each pair.
We need 6.
1 and 5 give 6.
With the signs: −1 and −5.
The two numbers go in the brackets: y = (x − 1)(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 − 1 is 0 when x = 1.
x − 5 is 0 when x = 5.
Careful!
Not −1.
x − 1 is 0 when x is 1: the sign in the bracket is the opposite.
So it crosses the x axis at x = 1 and x = 5: the points (1, 0) and (5, 0).
4 Show it on the graph
Here is the graph of y = x² − 6x + 5.
It crosses the x axis at 1 and 5.
5 Check
Let's check in the question's own form, y = x² − 6x + 5.
x = 1 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.