using quadratic formula show to find all real values of x such that the Square Root of (-3x^2 - x +24) is a complex number

Question

not sure what to do.

Answer 1

A square root is complex if it's applied to a negative number. So we need to find out when the quadratic <0.

-3x²-x+24<0, 3x²+x-24>0, (3x-8)(x+3)>0. This is true if both factors are positive or both factors are negative. So:

Positive: 3x>8, x>8/3,  x>-3. Since 8/3>-3 we choose x>8/3;

Negative: 3x<8, x<8/3, x<-3. Since -3<8/3 we choose x<-3. Therefore the square root is complex if x>8/3 or x<-3.