Step by step
The question
Question: solve 3x² + 4x − 2 = 0.
There is an x².
1 Try factoring
First: can it split into two brackets?
As in factoring, we need two numbers that multiply to a × c: 3 × −2 = −6.
They must add to 4.
We list the pairs that make 6.
One is minus, so we take the smaller from the bigger.
None gives 4.
So it does not split into brackets with whole numbers.
2 Find a, b and c
So we use the formula.
It works for every equation ax² + bx + c = 0, with 0 on one side.
First we read the numbers.
a = 3 is with x².
b = 4 is with x.
c = −2 is the plain number.
3 Use the formula
Here is the formula.
People worked it out long ago, and the lesson on completing the square shows where it comes from.
± means we do it twice: once with plus, once with minus.
We put the numbers in, each in brackets.
4 squared is 16.
4 times 3 times −2 is −24.
16 take away −24: taking away a minus number is the same as adding.
16 + 24 is 40.
So under the root we have 40.
40 is not a square number.
40 is 4 times 10, and √4 is 2.
So √40 is 2√10.
−4, 2 and the bottom 6: 2 goes into 4: 2 × 2 = 4.
2 goes into 6: 2 × 3 = 6.
So 2 goes into all of them.
We divide each by 2.
−4 divided by 2 is −2.
2√10 divided by 2 is √10.
The bottom becomes 3.
So x = (−2 ± √10)/3.
So x = (−2 + √10)/3 or x = (−2 − √10)/3.
With a calculator: x is about 0.39 or about −1.72.
Let's look back.
First: try factoring.
Then: find a, b and c.
Last: use the formula.