Question

How do you factor `4x^2-5x+7`?

Answer 1

It does not factorise

Explanation:

The rule to factorise any quadratic is to find two numbers such that

`"product" = x^2 " coefficient "xx" constant coefficient"`
`"sum" \ \ \ \ \ \ = x " coefficient"`

So for `4x^2-5x+7` we seek two numbers such that

`"product" = 4*17 = 28`
`"sum" \ \ \ \ \ \ = -5`

So if we looks at the factors of `28` and compute their sum we get (as the sum is negative and the product is positive then both factors must be negative);

`{: ("factor1", "factor2", "sum"), (-28,-1,-29), (-14,-2,-16), (-7,-4,-11) :}`

So as you can see we cannot find two such factors, and so we conclude the expression cannot be factorised

This approach works for all quadratics (assuming it does factorise) , The middle step in the last section can usually be skipped with practice.