Intersection of Line and Circle or Parabola

[silent]

Basically I have to find if the line intersects the circle in 2 points, one point, or none. Here is the problem with my work up to the point

2 Equations:

y = x^2 + 3x -1
4x - y = -1

My work so far, I plug the first equation into the y of the second equation and try to solve for x:

4x - x^2 + 3x - 1 = -1
Which I simplified to:
-x^2 + 7x
To:
x(-x + 7)

At this point x(-x + 7) I need to find out my x coordinates and then I can take them and plug them into the equations above and get the y coordinates? I am just unsure how to get the x coordinates at this point and then which equation do I plug them into to solve for y?

Thanks for the help.

 

[DrPhil]

Basically I have to find if the line intersects the circle in 2 points, one point, or none. Here is the problem with my work up to the point

2 Equations:

y = x^2 + 3x -1
4x - y = -1

My work so far, I plug the first equation into the y of the second equation and try to solve for x:

4x - (x^2 + 3x - 1) = -1...All 3 terms of y are multiplied by -1
Which I simplified to:
-x^2 + 7x
To:
x(-x + 7)

At this point x(-x + 7) I need to find out my x coordinates and then I can take them and plug them into the equations above and get the y coordinates? I am just unsure how to get the x coordinates at this point and then which equation do I plug them into to solve for y?

Thanks for the help.

Re-work with proper signs in y. You WILL get a quadratic equation to either factor or solve by the quadratic formula. You can tell how many roots there will be (0, 1, or 2) by looking at the 'discriminate" of the equation.

Once you have the possible values of x, plug x into either equation to find y (plugging into BOTH equations is a good way to check your work!). I don't see that you were asked to find the actual points, but perhaps just how many there are.

 

[HallsofIvy]

Basically I have to find if the line intersects the circle in 2 points, one point, or none. Here is the problem with my work up to the point

2 Equations:

y = x^2 + 3x -1
4x - y = -1

My work so far, I plug the first equation into the y of the second equation and try to solve for x:

Another way is to solve the second equation for y= 4x+ 1 and replace y in the first equation: 4x+ 1= x^2+ 3x- 1 which, after subtracting 4x+ 1 from both sides becomes x^2- x- 2= 0

4x - x^2 + 3x - 1 = -1
Which I simplified to:
-x^2 + 7x

No. You have 4x- y= 4x- (x^2+ 3x-1)= 4x- x^2- 3x+ 1= -1

To:
x(-x + 7)

At this point x(-x + 7) I need to find out my x coordinates and then I can take them and plug them into the equations above and get the y coordinates? I am just unsure how to get the x coordinates at this point and then which equation do I plug them into to solve for y?

Thanks for the help.