Question

How do you factor `y ^2 - 4y + 4`?

Answer 1

`y^2-4y+4 = (y-2)^2`

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 `y^2-4y-4` we seek two numbers such that

`"product" = (1)*4) = 4`
`"sum" \ \ \ \ \ \ = -4`

So we look at the factors of `4`. As the sum is negative and the product is positive then both of the factors must be negative, We can check every combination of the product factors:

`{: ("factor1", "factor2", "sum"), (4,1,5), (2,2,4), (-4,-1,-5), (-2,-2,-4) :}`

So the factors we seek are `color(blue)(-2)` and `color(green)(-2)`

Therefore we can factorise the quadratic as follows:

`\ \ \ \ \ y^2-4y-4 = y^2 color(blue)(-2)y + color(green)(-2)y +4`
`:. y^2-4y+4 = y(y-2) -2y(y -2)`
`:. y^2-4y+4 = (y-2)(y-2)`
`:. y^2-4y+4 = (y-2)^2`

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

An important quadratic relationship to be familiar with is;

`(a+b)^2 = a^2 +2ab +b^2`

With this we can spot intermediately that `y^2-4y+4` is a perfect square