Determine the value of k in y=kx^2-5x+2 hat 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)
To determine the value of k that results in the desired intersections, we need to set the quadratic equation equal to the equation of the line and solve for x. Let's go through each case:
a) Two points of intersection:
In this case, the quadratic equation and the line intersect at two points, so they have two common solutions. To find this, we’ll equate them and solve for x.
y = kx^2 - 5x + 2
y = -3x + 4
Setting these two equations equal to each other:
kx^2 - 5x + 2 = -3x + 4
Now, we need to solve this equation for x. Rearrange the equation to set it equal to zero:
kx^2 - 5x + 2 + 3x - 4 = 0
kx^2 - 2x - 2 = 0
At this point, we have a quadratic equation, and for it to have two solutions, the discriminant (b^2 - 4ac) must be greater than zero.
The discriminant formula is:
b^2 - 4ac
In our equation, a = k, b = -2, c = -2.
So, the discriminant is:
(-2)^2 - 4(k)(-2)
Simplifying:
4 + 8k
For two points of intersection, this discriminant should be greater than zero. So:
4 + 8k > 0
Now, solve this inequality for k:
8k > -4
k > -1/2
Therefore, any value above -1/2 for k will result in two points of intersection.
b) One point of intersection:
In this case, the quadratic equation and the line intersect at only one point, so they have one common solution. We follow the same steps as before.
Setting the equations equal to each other:
kx^2 - 5x + 2 = -3x + 4
Now, rearrange the equation:
kx^2 - 2x - 2 = 0
For one point of intersection, the discriminant should be equal to zero. So:
4 + 8k = 0
Solving this equation for k:
8k = -4
k = -1/2
Any value of k equal to -1/2 will result in one point of intersection.
c) No point of intersection:
In this case, the quadratic equation and the line do not intersect at any point. We equate the equations and solve for x, just like before.
Setting the equations equal to each other:
kx^2 - 5x + 2 = -3x + 4
Now, rearrange the equation:
kx^2 - 2x - 2 = 0
For no point of intersection, the discriminant should be less than zero. So:
4 + 8k < 0
Solving the inequality for k:
8k < -4
k < -1/2
Any value of k less than -1/2 will result in no point of intersection.