Determine the value of k in y=kx^2-5x+2 that will result in the intersection of the line y=-3x+4 with the quadratic at
a) two points (1 mark)
b) one points (1 mark)
c) no point (1 mark)
Please and thank you!
To determine the value of k in the quadratic equation y = kx^2 - 5x + 2 that results in the intersection with the line y = -3x + 4, we need to equate the two equations and solve for x.
a) For two points of intersect:
To find the value of k that would result in two intersection points between the quadratic and the line, we equate the two equations:
kx^2 - 5x + 2 = -3x + 4
Rearranging the equation, we get:
kx^2 - 5x + 3x - 2 = 0
Combining like terms:
kx^2 - 2x - 2 = 0
For the quadratic equation to have two distinct solutions, the discriminant (b^2 - 4ac) must be greater than zero. In this case, the discriminant formula is:
D = (-2)^2 - 4(k)(-2)
Setting the discriminant greater than zero:
(-2)^2 - 4(k)(-2) > 0
4 + 8k > 0
Solving the inequality for k:
8k > -4
k > -4/8
k > -1/2
Therefore, any value of k greater than -1/2 will result in the quadratic equation intersecting the line at two distinct points.
b) For one point of intersection:
To find the value of k that would result in one intersection point between the quadratic and the line, we have to find the discriminant D = 0.
Setting the discriminant equal to zero, we have:
4 + 8k = 0
8k = -4
k = -4/8
k = -1/2
Therefore, the value of k equal to -1/2 will result in the quadratic equation intersecting the line at one point.
c) For no intersection points:
To find the value of k that would result in no intersection points between the quadratic and the line, the discriminant must be less than zero.
Setting the discriminant less than zero:
4 + 8k < 0
8k < -4
k < -4/8
k < -1/2
Therefore, any value of k less than -1/2 will result in the quadratic equation having no intersection points with the line.
To summarize:
a) Two points of intersection: k > -1/2
b) One point of intersection: k = -1/2
c) No intersection point: k < -1/2