Question 128657
 From the quadratic formula

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}


the discriminant consists of all of the terms in the square root. So the discriminant is


{{{D=b^2-4ac}}}


the discriminant tells us how many solutions (and what type of solutions) we can expect for any quadratic.



Now let's find the discriminant for {{{y=x^2+x+3}}} (notice how {{{a=1}}}, {{{b=1}}} and {{{c=3}}}):


{{{D=b^2-4ac}}} Start with the given equation


{{{D=(1)^2-4*1*3}}} Plug in a=1, b=1, c=3


{{{D=1-4*1*3}}} Square 1 to get 1


{{{D=1-12}}} Multiply -4*1*3 to get -12


{{{D=-11}}} Combine 1 and -12 to get -11



Since the discriminant equals -11  (which is less than zero) , this means  there are two complex solutions.



Notice if we graph {{{y=x^2+x+3}}}, we get


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+x+3) }}}


and we can see that there are two complex solutions. So this visually verifies our answer.