Question

How do you solve `x^ { 2} + x + 14= 9/ 16`?

Answer 1

`x = -1/2+-sqrt(211)/4 i`

Explanation:

The difference of squares identity can be written:

`A^2-B^2 = (A-B)(A+B)`

We will use this with `A=(4x+2)` and `B=sqrt(211)i` where `i` is the imaginary unit satisfying `i^2=-1` below.

Given:

`x^2+x+14 = 9/16`

Multiply both sides by `16` to get:

`16x^2+16x+224 = 9`

Transpose and subtract `9` from both sides to get:

`0 = 16x^2+16x+215`

`color(white)(0) = (4x)^2+2(4x)(2)+2^2+211`

`color(white)(0) = (4x+2)^2+(sqrt(211))^2`

`color(white)(0) = (4x+2)^2-(sqrt(211)i)^2`

`color(white)(0) = (4x+2-sqrt(211)i)(4x+2+sqrt(211)i)`

Hence:

`4x = -2+-sqrt(211)i`

So:

`x = -1/2+-sqrt(211)/4 i`