Question

How do you solve `x^2 - 4x + 7 =0` by completing the square?

Answer 1

`x = 2+-sqrt(3)i`

Explanation:

Given:

`x^2-4x+7=0`

While completing the square we will find that this takes the form of the sum of a square and a positive number. As a result it has no solution in real numbers, but we can solve it using complex numbers.

The imaginary unit `i` satisfies `i^2=-1`

The difference of squares identity can be written:

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

We can use this with `a=(x-2)` and `b=sqrt(3)i` as follows:

`0 = x^2-4x+7`

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

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

`color(white)(0) = (x-2)^2-(sqrt(3)i)^2`

`color(white)(0) = ((x-2)-sqrt(3)i)((x-2)+sqrt(3)i)`

`color(white)(0) = (x-2-sqrt(3)i)(x-2+sqrt(3)i)`

Hence the two roots are:

`x = 2+sqrt(3)i" "` and `" "x = 2-sqrt(3)i`