Proving No Intersection: y=2x-1 & y=x^4+3x^2+2x

In summary, the problem is to prove that the line with equation y=2x-1 does not intersect the curve with equation y=x^4 + 3x^2 +2x. This can be solved using indirect proof by assuming that the equations do intersect and then finding a contradiction. By substituting x^2 = t and solving the resulting quadratic equation, it is shown that there is no real solution for the equation x^4+3x^2+1=0, therefore the lines do not intersect.
  • #1
msimard8
57
0
Heres the quesion

Prove that the line whose equation is y=2x-1 does not intersect the curve with equation y=x^4 + 3x^2 +2x.

We are suppose to solve this using indirect proof, thus assuming the equations do intersect, and proving that wrong.

i let the y's equal each other, but that isn't getting me anywhere

where should i start.

thanks
 
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  • #2
msimard8 said:
Heres the quesion

Prove that the line whose equation is y=2x-1 does not intersect the curve with equation y=x^4 + 3x^2 +2x.

We are suppose to solve this using indirect proof, thus assuming the equations do intersect, and proving that wrong.

i let the y's equal each other, but that isn't getting me anywhere

where should i start.

thanks

Of course that is getting you somewhere. Ever thought of substituting x^2 = t ?
 
  • #3
Let [itex]y_1(x) = 2x-1[/itex] and [itex]y_2(x) = x^4 + 3x^2 + 2x[/itex]. Look at [itex]f(x) = y_2(x) - y_1(x) = x^4 + 3x^2 + 1[/itex].

If [itex]y_1[/itex] and [itex]y_2[/itex] intersect at [itex]x_0 \in \mathbb{R}[/itex] then [itex]f(x_0) = 0[/itex]. Can you get the contradiction (what do you know about [itex]x^2, \, x^4[/itex] when [itex]x \in \mathbb{R}[/itex]?)?
 
  • #4
radou said:
Of course that is getting you somewhere. Ever thought of substituting x^2 = t ?


umm yea i got the roots, x=1 or x=1 or x=-1

but what does that mean
 
  • #5
msimard8 said:
umm yea i got the roots, x=1 or x=1 or x=-1

but what does that mean

How did you get these roots? Let's start again. Intersection means setting x^4+3x^2+2x = 2x - 1, which implies x^4+3x^2+1=0. Now, as said, substitute x^2 = t, and solve the quadratic equation. Both solutions of this equation t1 and t2 are negative. So, substituting back to x^2 = t means that there is no real solution for the equation x^4+3x^2+1=0, i.e. y1 = x^4+3x^2+2x and y2 = 2x - 1 don't intersect.
 
  • #6
thank you so much


i just made a simple sign error in factoring t^2 + t +1

which gave me wrong roots

thanks again
 

1. What does it mean for two lines to have no intersection?

When two lines have no intersection, it means that they do not share a common point or do not cross each other at any point. In other words, their graphs do not intersect.

2. How can you prove that the lines y=2x-1 and y=x^4+3x^2+2x have no intersection?

To prove that two lines have no intersection, you can compare their equations and see if there is a solution that satisfies both equations. In this case, you can set the equations equal to each other and solve for x. If there is no solution, then the lines have no intersection.

3. What is the relationship between the slopes of the lines y=2x-1 and y=x^4+3x^2+2x?

The slopes of the two lines are different. The slope of y=2x-1 is 2, while the slope of y=x^4+3x^2+2x is a much larger number, which can be seen by comparing the coefficients of the x^4 terms.

4. Can you use a graph to prove that the lines have no intersection?

Yes, you can use a graph to prove that two lines have no intersection. When graphed, the two lines will either not intersect at all or will intersect at only one point. If there is only one point of intersection, this means that the lines are actually the same line and therefore have infinite intersections. If there is no point of intersection, then the lines have no intersection.

5. What is the significance of proving that two lines have no intersection?

Proving that two lines have no intersection is important in many areas of science and mathematics. It can help determine if a system of equations has a unique solution, and it can also be used to find the point of intersection between two curves or lines. Additionally, it is a fundamental concept in geometry and can be applied in real-world scenarios, such as determining if two paths or roads will ever cross.

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