Step by step
The question
Question: solve x² + 2x − 15 = 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 −15, and they add to 2.
Look at the sign in front of 15: it is minus.
So one number is plus and one is minus.
Now the sign in front of 2x: it is plus.
So the bigger one is plus.
When we add a plus and a minus, the bigger one wins.
We list the pairs that make 15, smallest first.
For each pair, we check two things: they multiply to 15, and the bigger take away the smaller is 2.
One is minus, so adding them is a take away.
1 × 15 = 15.
15 − 1 = 14.
Not 2.
3 × 5 = 15.
5 − 3 = 2.
Both work!
So the two numbers are 3 and 5.
With the signs: −3 and 5.
So it is (x − 3)(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 − 3 = 0, we add 3 on both sides.
In x + 5 = 0, we take away 5 on both sides.
So x = 3 or x = −5.
3 Check
Let's check: put x = 3 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 = 3 or x = −5.
That is choice B.
5 Why not the others
A is −15 and 1.
Wrong: −15 × 1 = −15, but −15 + 1 = −14, not 2.
C is −3 and 5.
Wrong: signs swapped: (x + 3)(x − 5) gives −2x.
D is 3 and 5.
Wrong: 3 × 5 = 15, not −15.
Let's look back.
First: factor.
Then: one bracket is 0.
Then: check.
Then: pick the choice.
Last: why not the others.