Step by step
The question
Question: solve x² − 7x + 10 = 0.
There is an x².
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Factor
Factoring goes backwards.
We need two numbers.
They multiply to 10, and they add to −7.
Look at the sign in front of 10: it is plus.
So the two numbers have the same sign.
Now the sign in front of 7x: it is minus.
So both are minus.
We list the pairs that make 10, smallest first.
For each pair, we check two things: they multiply to 10, and they add to 7.
1 × 10 = 10.
1 + 10 = 11.
Not 7.
2 × 5 = 10.
2 + 5 = 7.
Both work!
So the two numbers are 2 and 5.
With the signs: −2 and −5.
So it is (x − 2)(x − 5) = 0.
2 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In x − 2 = 0, we add 2 on both sides.
In x − 5 = 0, we add 5 on both sides.
So x = 2 or x = 5.
3 Check
Let's check: put x = 2 into the left side.
We get 0, the same as the right side.
It works!
Now put x = 5 into the left side.
We get 0 again, the same as the right side.
So both answers are right!
4 Pick the choice
Our answer is x = 2 or x = 5.
That is choice C.
5 Why not the others
A is −5 and −2.
Wrong: (x + 2)(x + 5) gives +7x, not −7x.
B is 1 and 10.
Wrong: 1 × 10 = 10, but 1 + 10 = 11, not 7.
D is 3 and 4.
Wrong: 3 + 4 = 7, but 3 × 4 = 12, not 10.
Let's look back.
First: factor.
Then: one bracket is 0.
Then: check.
Then: pick the choice.
Last: why not the others.