Question 116197
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.


So let's find the discriminant for {{{x^2-6x+c}}}


{{{D=-6^2-4*1*c}}} Plug in a=1, b=-6


{{{D=36-4*1*c}}} Square -6 to get 36


{{{D=36-4c}}} Multiply -4*1*c to get -4c


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Now if the discriminant is greater than zero, then we'll have 2 real solutions.


{{{36-4c>0}}} Set the discriminant greater than zero



{{{-4c>0-36}}}Subtract 36 from both sides



{{{-4c>-36}}} Combine like terms on the right side



{{{c<(-36)/(-4)}}} Divide both sides by -4 to isolate c  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{c<9}}} Divide


So when {{{c<9}}}, we'll have two real solutions.



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Now when the discriminant {{{36-4c}}} is equal to zero, then we'll have one real solution:


{{{36-4c=0}}} Set the discriminant equal to zero



{{{-4c=0-36}}}Subtract 36 from both sides



{{{-4c=-36}}} Combine like terms on the right side



{{{c=(-36)/(-4)}}} Divide both sides by -4 to isolate c




{{{c=9}}} Divide


So when {{{c=9}}}, we'll have one real solution.




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Now when the discriminant {{{36-4c}}} is less than zero, then we'll have no real solutions:



{{{36-4c<0}}} Set the discriminant less than zero



{{{-4c<0-36}}}Subtract 36 from both sides



{{{-4c<-36}}} Combine like terms on the right side



{{{c>(-36)/(-4)}}} Divide both sides by -4 to isolate c  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{c>9}}} Divide



So when {{{c>9}}}, we won't have any real solutions