Question

How do you use the quadratic formula to solve the equation, `x^2-x= -1`?

Answer 1

NO ROOTS in `x !in RR`
ROOTS `x in CC`
`x=(1+isqrt3)/2`
OR
`x=(1-isqrt3)/2`

Explanation:

`x^2-x=-1`
`rArrx^2-x+1=0`
We have to factorize
`color(brown)(x^2-x+1)`
Since we can not use polynomial identities so we will calculate `color(blue)(delta)`

`color(blue)(delta=b^2-4ac)`
`delta=(-1)^2-4(1)(1)=-3<0`

NO ROOTS IN `color(red)(x !in RR)` because `color(red)(delta<0)`

But roots exist in `CC`
`color(blue)(delta=3i^2)`

Roots are
`x_1=(-b+sqrtdelta)/(2a)=(1+sqrt(3i^2))/2=(1+isqrt3)/2`
`x_2=(-b-sqrtdelta)/(2a)=(1-sqrt(3i^2))/2=(1-isqrt3)/2`

The equation is:
`x^2-x+1=0`
`rArr(x-(1+isqrt3)/2)(x-(1-isqrt3)/2)=0`
`(x-(1+isqrt3)/2)=0rArrcolor(brown)( x=(1+isqrt3)/2)`
OR
`(x-(1-isqrt3)/2)=0rArrcolor(brown)(x=(1-isqrt3)/2)`

So the roots exist only in `color(red)(x in CC)`