Step by step
The question
Question: solve x² + 3x − 28 = 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 −28, and they add to 3.
Look at the sign in front of 28: it is minus.
So one number is plus and one is minus.
Now the sign in front of 3x: 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 28, smallest first.
For each pair, we check two things: they multiply to 28, and the bigger take away the smaller is 3.
One is minus, so adding them is a take away.
1 × 28 = 28.
28 − 1 = 27.
Not 3.
2 × 14 = 28.
14 − 2 = 12.
Not 3.
4 × 7 = 28.
7 − 4 = 3.
Both work!
So the two numbers are 4 and 7.
With the signs: −4 and 7.
So it is (x − 4)(x + 7) = 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 − 4 = 0, we add 4 on both sides.
In x + 7 = 0, we take away 7 on both sides.
So x = 4 or x = −7.
3 Check
Let's check: put x = 4 into the left side.
We get 0, the same as the right side.
It works!
Now put x = −7 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 = 4 or x = −7.
That is choice B.
5 Why not the others
A is −14 and 2.
Wrong: −14 × 2 = −28, but −14 + 2 is −12, not 3.
C is −4 and 7.
Wrong: signs swapped: (x − 7)(x + 4) gives −3x.
D is 4 and 7.
Wrong: 4 × 7 = 28, but we need −28.
Let's look back.
First: factor.
Then: one bracket is 0.
Then: check.
Then: pick the choice.
Last: why not the others.