Step by step
The question
Question: solve 2x² − 5x − 4 = 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: 2 × −4 = −8.
They must add to −5.
We list the pairs that make 8.
One is minus, so we take the smaller from the bigger.
None gives 5.
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 = 2 is with x².
b = −5 is with x.
c = −4 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.
−5 squared is 25.
4 times 2 times −4 is −32.
25 take away −32: taking away a minus number is the same as adding.
25 + 32 is 57.
So under the root we have 57.
57 is not a square number.
We keep √57.
So x = (5 + √57)/4 or x = (5 − √57)/4.
With a calculator: x is about 3.14 or about −0.64.
Let's look back.
First: try factoring.
Then: find a, b and c.
Last: use the formula.