Question

How do you solve the equation `2x^2-3x+1=0` by completing the square?

Answer 1

`x=1" "` or `" "x=1/2`

Explanation:

`f(x) = 2x^2-3x+1`

The difference of squares identity can be written:

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

We will use this with `a=(4x-3)` and `b=1`.

First pre-multiply by `8` to avoid the need to do arithmetic with fractions:

`0 = 8f(x)`

`color(white)(0) = 8(2x^2-3x+1)`

`color(white)(0) = 16x^2-24x+8`

`color(white)(0) = (4x)^2-2(4x)(3)+9-1`

`color(white)(0) = (4x-3)^2-1^2`

`color(white)(0) = ((4x-3)-1)((4x-3)+1)`

`color(white)(0) = (4x-4)(4x-2)`

`color(white)(0) = (4(x-1))(2(2x-1))`

`color(white)(0) = 8(x-1)(2x-1)`

So `x=1` or `x=1/2`