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Solve the system of equations using any method.

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We have y is equal to 2 times the quantity x minus 4

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squared plus 3.

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We also have y is equal to negative x squared

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plus 2 x minus 2.

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The solution-- it might be one, it might be none, or it

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might be two solutions-- to this system occurs for the x

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values that generate the same y values.

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There's the same x and y that satisfy

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both of these equations.

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In order to find the x values, they need to equal the same y

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values, so this y has to be that y value.

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So the solution is going to occur when this guy right

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here-- negative x squared plus 2x minus 2 is equal to that

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guy up there, or equal to 2 times x minus 4

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squared plus 3.

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Now let's just try to solve for x.

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The left hand side-- we're going to have to multiply this

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out, so let's do that first. It's negative x squared plus

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2x minus 2 is equal to.

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And on the right hand side, 2 times x minus 4 squared is x

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squared minus 8x plus 16 plus 3.

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This is going to be equal to 2x squared-- I'm just

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distributing the 2-- minus 16x plus 32 plus 3, which is equal

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to 2x squared minus 16x plus 35.

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That's, of course, going to be equal to this thing on the

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left hand side, negative x squared plus 2x minus 2.

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Let's just get rid of this whole thing from the left hand

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side all at once by adding x squared to both sides.

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We can all do it in one step.

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We're going to add x squared to both sides.

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Let's subtract 2x from both sides, and let's

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add 2 to both sides.

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On the left hand side, those cancel out, those cancel out,

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those cancel out.

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You're left with 0 is equal to 2x squared plus x squared is

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3x squared.

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Negative 16x minus 2x is negative 18x, and then

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35 plus 2 is 37.

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So we just have a plain vanilla quadratic equation

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right here.

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We might as well apply the quadratic formula here to try

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to solve it.

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Our solutions are going to be x is equal to negative b.

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Well, b is negative 18, so negative b is positive 18.

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It's 18 plus or minus the square root of 18 squared

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minus 4 times 3 times c-- times 37.

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All of that is over 2 times a-- 2 times 3, which is 6.

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Let's think about what this is going to be.

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Over here, we have 18 plus or minus the square root of--

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let's just use a calculator.

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I could multiply it out but I think-- we have 18 squared

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minus 4 times 3 times 37, which is negative 120.

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It's 18 plus or minus the square root of negative 120.

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You might have even been able to figure out

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that this is negative.

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4 times 3 is 12.

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12 times 37 is going to be a bigger number than 18.

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Although it's not 100% obvious, but you might be able

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to just get the intuition there.

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We definitely end up with a negative number under the

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radical here.

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Now, if we're dealing with real numbers, there is no

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square root of negative 120.

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So there is no solution to this quadratic equation.

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There is no solution.

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If we wanted to, we could have just looked at the

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discriminant.

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The discriminant is this part-- b squared minus 4ac.

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We see the discriminant is negative, there's no solution,

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which means that these two guys-- these two equations--

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never intersect.

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There is no solution to the system.

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There are no x values that when you put into both of

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these equations give you the exact same y value.

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Let's think a little bit about why that happened.

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This one is already in kind of our y-intercept form.

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It's an upward opening parabola, so it looks

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something like this.

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I'll do my best to draw it-- just a quick and

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dirty version of it.

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Let me draw my axes in a neutral color.

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Let's say that this right here is my y-axis, that right there

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is my x-axis.

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x and y.

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This vertex-- it's in the vertex form-- occurs when x is

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equal to 4 and y is equal to 3.

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So x is equal to 4 and y is equal to 3.

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It's an upward opening parabola.

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We have a positive coefficient out here.

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So this will look something like this.

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I don't know the exact thing, but that's close enough.

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Now, what will this thing look like?

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It's a downward opening parabola and we can actually

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put this in vertex form.

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Let me put the second equation in vertex form,

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just so we have it.

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So we have a good sense.

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So, y is equal to-- we could factor in a negative 1--

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negative x squared minus 2x plus 2.

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Actually, let me put the plus 2 further out-- plus 2, all

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the way up out there.

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Then we could say, half of negative 2 is negative 1.

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You square it, so you have a plus 1 and

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then a minus 1 there.

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This part right over here, we can rewrite as x minus 1

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squared, so it becomes negative x minus 1 squared.

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Let me just do it one step at a time.

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I don't want to skip steps.

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Negative x minus 1 squared minus 1 plus 2.

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So that's plus 1 out here.

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Or if we want to distribute the negative, we get y is

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equal to negative x minus 1 squared minus 1.

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Here the vertex occurs at x is equal to 1, y is equal to

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negative 1.

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The vertex is there, and this is a

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downward opening parabola.

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We have a negative coefficient out here on the second degree

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term, so it's going to look something like this.

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So as you see, they don't intersect.

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This vertex is above it and it opens upward.

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This is its minimum point.

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And it's above this guy's maximum point.

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So they will never intersect, so there is no solution to

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this system of equations.

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