for what values of k in y=3x+k with the curve y=-x^2-3x+6 intersect the line at 2 points
let's intersect them:
-x^2 - 3x + 6 = 3x + k
-x^2 - 6x + 6-k = 0
x^2 + 6x + k-6 = 0
to have 2 distinct real solutions , the discriminant must be > 0
b^2 - 4ac > 0
36 - 4(1)(k-6) > 0
36 -4k + 24 > 0 > 0
-4k > -60
k < 15
Notice in my graph, when k = 0, the straight line is a tangent, so for k< 15, the line would drop down and you would have 2 intersection points
http://www.wolframalpha.com/input/?i=plot+y+%3D+-x%5E2+-+3x+%2B+6+%2C+y+%3D+3x+%2B+15
To find the values of k for which the curve y = 3x + k intersects the given curve y = -x^2 - 3x + 6 at two points, we need to find the discriminant of the quadratic equation formed by setting the two curves equal to each other.
Step 1: Set the equations equal to each other:
3x + k = -x^2 - 3x + 6
Step 2: Rearrange the equation to standard quadratic form:
x^2 - 6x + (k - 6) = 0
Step 3: Find the discriminant:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
In our equation, a = 1, b = -6, and c = (k - 6).
D = (-6)^2 - 4(1)(k - 6)
D = 36 - 4(k - 6)
D = 36 - 4k + 24
D = -4k + 60
Step 4: Determine the conditions for two points of intersection:
For the quadratic equation to have two distinct real roots (and hence, two points of intersection), the discriminant must be greater than zero.
D > 0
-4k + 60 > 0
Step 5: Solve the inequality:
-4k + 60 > 0
-4k > -60
k < 15
Therefore, the values of k for which the curve y = 3x + k intersects the curve y = -x^2 - 3x + 6 at two points are k < 15.
To find the values of k for which the curve intersects the line at two points, we need to determine the discriminant of the quadratic equation.
The given equation of the curve is y = -x^2 - 3x + 6, and the equation of the line is y = 3x + k.
To find the intersection points of these two equations, we can set them equal to each other:
-x^2 - 3x + 6 = 3x + k
Rearranging this equation to standard form:
-x^2 - 6x + 6 - 3x - k = 0
Now we can compute the discriminant using the general quadratic equation formula: Δ = b^2 - 4ac.
In this case, a = -1, b = -6, and c = 6 - k. Substituting the values into the formula:
Δ = (-6)^2 - 4(-1)(6 - k)
= 36 + 4(6 - k)
= 36 + 24 - 4k
= 60 - 4k
For the quadratic equation to have two distinct real roots (two intersection points), the discriminant needs to be greater than zero. So we have:
60 - 4k > 0
Solving this inequality for k:
60 > 4k
k < 15
Therefore, for values of k less than 15, the curve y = -x^2 - 3x + 6 intersects the line y = 3x + k at two points.