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CCSS.Math: ,

solve the system of equations using any method so we have y is equal to 2 times the quantity X minus 4 squared plus 3 we also have y is equal to negative x squared plus 2x minus 2 so the solution or if it might be one or might be none it might be two solutions but the solutions to this quadratic system occur for the X values that generate the same y value so the same X&Y that satisfy both of these equations so in order to find the x values they need to equal the same Y value so this Y has to be that Y value so the solution is going to occur when this guy right here negative x squared plus 2x minus 2 is equal to is equal to that guy up there is equal to 2 times X minus 4 squared plus 3 and now let's just try to solve for X so the left-hand side not LAT well let's let's just well we're going to have to multiply this out so let's do that first so it's negative x squared plus 2x minus 2 is equal to and on the right hand side 2 times X minus 4 squared is x squared minus 8x plus 16 plus 3 this is going to be equal to 2x squared I'm just distributing the 2 minus 16x 2x squared minus 16x plus 32 plus 3 which is equal to this is equal to 2x squared minus 16x plus 35 and that's of course going to be equal to this thing on the left hand side negative x squared plus 2x minus 2 and now let's well let's just let's just subtract this let's just get rid of this whole thing from the left hand side all at once by adding x squared to both sides we can all do it in one step we're going to add x squared to both sides let's subtract 2x from both sides let's subtract 2x from both sides and let's add 2 to both sides and let's add 2 to both sides and we will get on the left hand side those cancel out those cancel out those cancel out you're left with 0 is equal to 2x squared plus X square Aird is 3x squared negative 16x minus 2x is negative 18x and then 35 plus 2 is 37 plus 37 so we just have a plain vanilla quadratic equation right here and we might as well apply the quadratic formula here to try to solve it so our solutions are going to be X is equal to negative B well B is negative 18 so negative B is positive 18 so it's 18 plus or minus the square root of 18 squared 18 squared minus 4 times 3 times 3 times C times 37 all of that over 2 times a 2 times 3 which is 6 now let's think about what this is going to be over here we have so it's 18 plus or minus the square root of well let's just use our calculator I could multiply it out but I think so if 18 squared 18 squared minus 4 times 3 times 37 negative 120 so 18 plus or minus square root of negative 120 and you might have even be able to figure out this is negative 4 times 3 is 12 12 times 37 is going to be a bigger number than 18 although it's you know it's not 100% obvious but you might be able to just get the intuition there but we definitely end up with a negative number under the radical here now if we're dealing with real numbers there is no square root of negative 120 so there is no solution to this quadratic equation there is no solution and if we wanted to we could have just looked at the discriminant the discriminant is this part b squared minus 4ac we see the discriminant is negative there's no solution which means that these two guys these two equations never intersect there is no solution to the system there are no x-values that when you put into both of these equations give you the exact same y-value now let's think a little bit why that happened this one is already in kind of our y-intercept form and it's an upward-opening parabola so it looks something like this I'll do my best to draw it just a quick and dirty version of it we draw my axes in a in a neutral color so let's say that this right here is my y-axis that right there is my x-axis x and y this vertex it's in the vertex form occurs it with X is equal to 4 and Y is equal to 3 so X is equal to 4 and Y is equal to 3 and it's an upward-opening parabola we have a positive coefficient out here so this will look something like this it will look something like this I don't know the exact thing but look that's close enough now what will this thing look like well it's a downward-opening parabola and we could actually put this in vertex form let me do that let me do that let me put the second equation in vertex form just so we have it so you have a good sense so Y is equal to we could factor out a negative 1 negative x squared minus 2x plus 2 x squared minus 2x plus 2 and actually let me put the plus 2 further out plus 2 all the way up out there and then we could say ok half of negative 2 is negative 1 you square it so you have a plus 1 and then a minus 1 there and then this part right over here we can rewrite as X minus 1 squared so it becomes negative X minus 1 squared let me just do it one step at a time I don't want to skip steps negative x minus 1 squared minus 1 plus 2 so that's plus 1 out here or if we want to distribute the negative we get Y is equal to negative x minus 1 squared minus 1 so here the vertex occurs at X is equal to 1 Y is equal to negative 1 X is equal to 1 Y is equal to negative 1 the vertex is there and this is a downward-opening parabola we have a negative coefficient out here on the on the second degree term so it's going to look something like this it's going to look something like this so as you see they don't intersect this is this vertex is above it and it opens upward this is its minimum point and it's above this guy's maximum point so they will never intersect so there is no solution to this system of equations