Non-linear systems of equations
Non-Linear Systems of Equations 3 Non-Linear Systems of Equations 3
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- Solve the system of equations using any method.
- 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 2 x minus 2.
- The solution-- it might be one, it might be none, or it
- might be two solutions-- to this system occurs for the x
- values that generate the same y values.
- There's the same x and y that satisfy
- both of these equations.
- In order to find the x values, they need to equal the same y
- values, 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 that
- guy up there, or equal to 2 times x minus 4
- squared plus 3.
- Now let's just try to solve for x.
- The left hand side-- we're going to have to multiply this
- out, so let's do that first. 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 plus 32 plus 3, which is equal
- to 2x squared minus 16x plus 35.
- That's, of course, going to be equal to this thing on the
- left hand side, negative x squared plus 2x minus 2.
- 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, and let's
- add 2 to both sides.
- 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 squared is
- 3x squared.
- Negative 16x minus 2x is negative 18x, and then
- 35 plus 2 is 37.
- So we just have a plain vanilla quadratic equation
- right here.
- We might as well apply the quadratic formula here to try
- to solve it.
- Our solutions are going to be x is equal to negative b.
- Well, b is negative 18, so negative b is positive 18.
- It's 18 plus or minus the square root of 18 squared
- minus 4 times 3 times c-- times 37.
- All of that is over 2 times a-- 2 times 3, which is 6.
- Let's think about what this is going to be.
- Over here, we have 18 plus or minus the square root of--
- let's just use a calculator.
- I could multiply it out but I think-- we have 18 squared
- minus 4 times 3 times 37, which is negative 120.
- It's 18 plus or minus the square root of negative 120.
- You might have even been able to figure out
- that this is negative.
- 4 times 3 is 12.
- 12 times 37 is going to be a bigger number than 18.
- Although it's not 100% obvious, but you might be able
- to just get the intuition there.
- 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.
- 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.
- Let's think a little bit about why that happened.
- This one is already in kind of our y-intercept form.
- 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.
- Let me draw my axes in a neutral color.
- 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 when x is
- equal to 4 and y is equal to 3.
- So x is equal to 4 and y is equal to 3.
- It's an upward opening parabola.
- We have a positive coefficient out here.
- So this will look something like this.
- I don't know the exact thing, but that's close enough.
- Now, what will this thing look like?
- It's a downward opening parabola and we can actually
- put this in vertex form.
- Let me put the second equation in vertex form,
- just so we have it.
- So we have a good sense.
- So, y is equal to-- we could factor in a negative 1--
- negative x squared minus 2x plus 2.
- Actually, let me put the plus 2 further out-- plus 2, all
- the way up out there.
- Then we could say, half of negative 2 is negative 1.
- You square it, so you have a plus 1 and
- then a minus 1 there.
- 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.
- Here the vertex occurs at 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 second degree
- term, so it's going to look something like this.
- So as you see, they don't intersect.
- 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.
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