Question 119529
I'll do the first two to help you get started



#22


 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^2-3x+2}}}:


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


{{{D=9-4*1*2}}} Square -3 to get 9


{{{D=9-8}}} Multiply -4*1*2 to get -8


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



Since the discriminant equals 1  (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.




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#24



 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=-3x^2+5x-1}}}:


{{{D=5^2-4*-3*-1}}} Plug in a=-3, b=5, c=-1


{{{D=25-4*-3*-1}}} Square 5 to get 25


{{{D=25-12}}} Multiply -4*-3*-1 to get -12


{{{D=13}}} Combine 25 and -12 to get 13



Since the discriminant equals 13  (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.