Question 118380
 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. If the discriminant is greater than zero, then we will have 2 real solutions. If the discriminant is equal to zero, then we will have one real solution. Finally, if the discriminant is less than zero, then we will have 2 imaginary solutions. 


So let's use the discriminant to find the number and type of solutions {{{y=8x^2+3}}} has:



{{{D=0^2-4*8*3}}} Plug in a=8, b=0, c=3


{{{D=0-4*8*3}}} Square 0 to get 0


{{{D=0-96}}} Multiply -4*8*3 to get -96


{{{D=-96}}} Combine 0 and -96 to get -96



Since the discriminant equals -96 (which is less than zero), this means there are two imaginary solutions




Notice if we graph {{{y=8x^2+3}}} (to graph, just plot a few points), we get


{{{ graph( 500, 500, -10, 10, -10, 10, 8x^2+3) }}}


and we can see that the graph does not cross the x-axis, so there are two imaginary solutions.