Find the values of m for which the line with equation y=mx+2 does not intersect the parabola with equation y=(x-1)^2 + 5.
I am not sure how to solve this problem. I know that both equations equal each other when it intersects but what happens if they don't intersect. Please help me.
if the line intersects, then we have
(x-1)^2+5 = mx+2
x^2-2x+6 = mx+2
x^2-(m+2)x+4 = 0
For no solutions, you must have the discriminant negative:
(m+2)^2 - 16 < 0
...
Hey there! Looks like you need some help with your math problem. Don't worry, I've got your back! Now, let's find the values of m for which the line and the parabola do not intersect.
To solve this, we can set the equations equal to each other and then find the discriminant. The discriminant will tell us if the line and parabola intersect (if it's greater than zero), or if they don't intersect (if it's less than zero).
So, let's set the two equations equal:
mx + 2 = (x-1)^2 + 5
Expanding and simplifying the equation, we get:
mx + 2 = x^2 - 2x + 1 + 5
Combining like terms, we get:
mx + 2 = x^2 - 2x + 6
Rearranging to standard form, we have:
x^2 - (m+2)x + 6 - 2 = 0
x^2 - (m+2)x + 4 = 0
Now, let's find the discriminant, which is given by:
Discriminant = (m+2)^2 - 4(1)(4)
Simplifying further, we get:
Discriminant = m^2 + 4m + 4 - 16
Discriminant = m^2 + 4m - 12
For the line and the parabola to not intersect, the discriminant should be less than zero. This is because for two curves to intersect, the discriminant should be greater than zero.
So, we have the inequality:
m^2 + 4m - 12 < 0
Solving this inequality, we get:
m < -6 or m > 2
Therefore, the values of m for which the line with equation y = mx + 2 does not intersect the parabola with equation y = (x-1)^2 + 5 are m < -6 or m > 2.
I hope that helps! Good luck with your math problem! If you have any more questions, feel free to ask.
To find the values of m for which the line y = mx + 2 does not intersect the parabola y = (x - 1)^2 + 5, we can set up a system of equations and solve for the intersection points.
Setting the equations equal to each other, we have:
mx + 2 = (x - 1)^2 + 5
Expanding the square on the right side:
mx + 2 = x^2 - 2x + 1 + 5
Rearranging and combining like terms:
x^2 - (m + 2)x + (8 - 1) = 0
Now, for the line y = mx + 2 to not intersect the parabola, the quadratic equation above should not have any real solutions. This means that the discriminant (b^2 - 4ac) should be less than zero.
The discriminant, in this case, is:
(b^2 - 4ac) = (m + 2)^2 - 4(1)(7) = m^2 + 4m - 24
Setting the discriminant less than zero:
m^2 + 4m - 24 < 0
Next, we solve this inequality by factoring or using the quadratic formula:
(m + 6)(m - 4) < 0
To determine the values of m that satisfy this inequality, we set each factor equal to zero and analyze the intervals between the solutions:
m + 6 = 0 → m = -6
m - 4 = 0 → m = 4
Using a number line representation, we can see that the values of m that make the inequality less than zero are between -6 and 4:
-6 4
|---------------|-----------------|
x x x
Therefore, the values of m for which the line y = mx + 2 does not intersect the parabola y = (x - 1)^2 + 5 are -6 < m < 4.
To find the values of m for which the line y = mx + 2 does not intersect the parabola y = (x - 1)^2 + 5, we need to compare the equations and analyze their relationship.
First, we note that the line and the parabola are both in the form of y = mx + b, where b is the y-intercept. In this case, the y-intercept of the line is 2, and the y-intercept of the parabola is 5.
To determine if the line and the parabola intersect, we need to set their equations equal to each other and solve for x and y simultaneously. In this case, we have:
mx + 2 = (x - 1)^2 + 5
Expanding the right side of the equation, we get:
mx + 2 = x^2 - 2x + 1 + 5
Rearranging the terms, we have:
x^2 - (m + 2)x + 8 = 0
Now, for the line and the parabola not to intersect, the above equation must not have any real solutions for x. This means that the discriminant of the quadratic equation, b^2 - 4ac, must be less than zero.
The discriminant, in this case, is:
(m + 2)^2 - 4(1)(8)
Simplifying further, we have:
m^2 + 4m + 4 - 32 = 0
m^2 + 4m - 28 = 0
Now, for there to be no real solutions, the discriminant of this quadratic equation must be less than zero. Therefore, we have:
4^2 - 4(1)(-28) < 0
16 + 112 < 0
Clearly, this inequality is false since 16 + 112 = 128, which is greater than zero.
So, to summarize, the line y = mx + 2 intersects the parabola y = (x - 1)^2 + 5 for all values of m. In other words, there are no values of m for which the line and the parabola do not intersect.