Question 116123
{{{y= -3x^2 + 6x - 2}}}.  By 'solve' I presume you mean to find the roots of the equation where y = 0.


Since this cannot be factored, you must either complete the square or use the quadratic formula {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}(which is really the same thing).


In this equation: {{{-3x^2 + 6x - 2=0}}}, {{{a=-3}}}, {{{b=6}}}, and {{{c=-2}}}


{{{x = (-6 +- sqrt( 6^2-4*(-3)*(-2) ))/(2*(-3)) }}} 


{{{x= (-6 +- sqrt(12))/(-6)}}}


{{{x=(-6+2sqrt(3))/-6}}} or {{{x=(-6-2sqrt(3))/-6}}}
{{{x=(3-sqrt(3))/3}}} or {{{x=(3+sqrt(3))/3}}}


To graph the equation, first find the x-coordinate of the vertex given by {{{-b/2a=-6/-6=1}}}, then evaluate the function f(x)=y at this x value:  {{{f(1)=-3+6-2=1}}}.  Therefore the vertex is at (1, 1).  Since the coefficient on the {{{x^2}}} term is negative, you know the graph is convex down.  With the vertex at (1,1), we have an axis of symmetry at the vertical line {{{x=1}}}.  {{{f(0)=0+0-2=-2}}}, so we have the y-axis intercept at (0,-2).  The roots of the equation that we calculated above come out to a little less than .5 for one and a little more than 1.5 for the other.  That's close enough for graphing purposes.  If you want to pick a few more values for x to give you some additional points to smooth the curve, pick them in the neighborhood of .5 and 1.5.  Plot all of these points and then draw a smooth curve.


{{{graph(600,600,-5,5,-5,5,-3x^2+6x-2)}}}


Super-Double-Plus Extra Credit:  Why did I say that using the quadratic formula and completing the square are really the same thing?


Hope that helps,
John