Step by step
The question
Question: which form of y = 2x² − 8x + 6 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
First we take 2 out of every part: each part divided by 2.
Then x squared has no number in front.
So y is 2 times (x² − 4x + 3).
Factoring goes backwards.
We need two numbers.
They multiply to 3, and they add to −4.
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 3, smallest first.
We add each pair.
We need 4.
1 and 3 give 4.
With the signs: −1 and −3.
The two numbers go in the brackets: y = 2(x − 1)(x − 3).
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.
The 2 in front is never 0.
x − 1 is 0 when x = 1.
x − 3 is 0 when x = 3.
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 = 3: the points (1, 0) and (3, 0).
4 Show it on the graph
Here is the graph of y = 2x² − 8x + 6.
It crosses the x axis at 1 and 3.
5 Check
Let's check in the question's own form, y = 2x² − 8x + 6.
x = 1 gives y = 0: on the x axis.
And x = 3 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.