Question 132421
 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=2x^2+x-1}}}:


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


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


{{{D=1-4*2*-1}}} Square 1 to get 1


{{{D=1+8}}} Multiply -4*2*-1 to get 8


{{{D=9}}} Combine 1 and 8 to get 9



Since the discriminant equals 9  (which is greater than zero) , this means  there are two real solutions. Remember if the discriminant is greater than zero, then the quadratic will have two real solutions.




So the quadratic {{{y=2x^2+x-1}}} has two solutions.




Notice if we graph {{{y=2x^2+x-1}}}, we can see that there are two real solutions. So this verifies our answer. 



{{{ graph( 500, 500, -10, 10, -10, 10, 2x^2+x-1) }}}