Question

How do you solve `x^2+18=-81`?

Answer 1

`x^2=-99=>x=+-sqrt(-99)=+-3sqrt(11)i`

Answer 2

`color(blue)(=>x=+-3isqrt(11))`

Explanation:

Given:`" "color(brown)(x^2+18=-81)`

If so chosen you could write this as `y=x^2+99`

`color(blue)("Solving for "x" where the given equation is true")`

Subtract `color(blue)(18)` from both sides

`" "color(brown)(x^2+18color(blue)(-18)=-81color(blue)(-18))`

`" "x^2+0=-99`

know: `color(green)(3xx33=99)" so "color(blue)(3xx3xx11=99)" so "color(red)(3^2xx11=99)`

But we have negative 99 so we have `(-1)xx3^2xx11`

Thus our equation becomes

`x^2=(-1)xx3^2xx11`

Take the square root of each side

`x=+-sqrt((-1)xx3^2xx11)`

`=> x=+-3xxsqrt(11)sqrt(-1)`

But `sqrt(-1)=i`

`color(blue)(=>x=+-3isqrt(11))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Notice that the graph does not cross the x-axis.

But by writing the equation as `x^2+18=-81`

`y=x^2+99`

So to solve for `x` we have to write

`y=0=x^2+99` but for `y=0` implies that the graph crosses the x-axis when in fact it does not. So to be able to solve for `y=` we introduce 'complex numbers'. Hence the `sqrt(-1)`