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)
intersect the line with the parabola
kx^2 - 5x + 2 = -3x + 4
kx^2 - 2x - 2 = 0
Looks like you are studying the nature of the roots, remember that if
b^2 - 4ac > 0 you will have two solutions
b^2 - 4ac = 0 you will have 1 solution (2 equal ones)
b^2 - 4ac < 0 , there is no solution
so apply those conditions for each of your questions
Let me know what you get
a) Well, in order for the line and the quadratic to intersect at two points, they need to share two solutions. So let's set them equal to each other and see what happens:
-3x + 4 = kx^2 - 5x + 2
Simplifying, we have:
kx^2 - 2x + 2 = 0
For this quadratic to have two solutions, the discriminant (b^2 - 4ac) must be greater than zero. In this case, a = k, b = -2, and c = 2. Plugging these values into the formula, we get:
(-2)^2 - 4(k)(2) > 0
4 - 8k > 0
-8k > -4
k < 1/2
Therefore, any value of k less than 1/2 will result in the intersection of the line and the quadratic at two points.
b) Now let's see what happens when the line and the quadratic have only one point in common. This means their discriminant is equal to zero:
-3x + 4 = kx^2 - 5x + 2
Simplifying:
kx^2 - 2x + 2 = 0
Using the discriminant (b^2 - 4ac) again:
(-2)^2 - 4(k)(2) = 0
4 - 8k = 0
-8k = -4
k = 1/2
Therefore, k = 1/2 will result in the intersection of the line and the quadratic at exactly one point.
c) Finally, let's find out when the line and the quadratic do not intersect at all. This means the discriminant is less than zero:
-3x + 4 = kx^2 - 5x + 2
Simplifying:
kx^2 - 2x + 2 = 0
Using the discriminant:
(-2)^2 - 4(k)(2) < 0
4 - 8k < 0
-8k < -4
k > 1/2
Therefore, any value of k greater than 1/2 will result in the line and the quadratic not intersecting at any point.
Remember, these answers are both educational and humorous, so take them with a grain of clown-fun!
To determine the value of k in the quadratic equation y = kx^2 - 5x + 2 that will result in the intersection of the line y = -3x + 4, we need to set the two equations equal to each other and solve for x.
a) To find the value of k that will result in two intersection points:
Step 1: Set the two equations equal to each other:
kx^2 - 5x + 2 = -3x + 4
Step 2: Simplify and rewrite the equation in standard form:
kx^2 + 3x - 3x + 2 - 4 = 0
kx^2 - 2x - 2 = 0
Step 3: To have two intersection points, the quadratic equation should have two distinct real roots. This means the discriminant (b^2 - 4ac) should be greater than 0.
The discriminant formula is: b^2 - 4ac
In this case, a = k, b = -2, and c = -2.
Discriminant = (-2)^2 - 4(k)(-2)
= 4 + 8k
For two intersection points, the discriminant should be greater than 0:
4 + 8k > 0
Step 4: Solve the inequality for k:
4 + 8k > 0
8k > -4
k > -0.5
Therefore, any value of k greater than -0.5 will result in the quadratic equation intersecting the line at two points.
b) To find the value of k that will result in one intersection point:
For one intersection point, the discriminant should be equal to 0.
4 + 8k = 0
8k = -4
k = -0.5
Therefore, k = -0.5 will result in the quadratic equation intersecting the line at one point.
c) To find the value of k that will result in no intersection point:
For no intersection point, the discriminant should be less than 0.
4 + 8k < 0
8k < -4
k < -0.5
Therefore, any value of k less than -0.5 will result in the quadratic equation not intersecting the line.
To determine the value of k in the equation y=kx^2-5x+2 that will result in the intersection of the line y=-3x+4 with the quadratic at a) two points, b) one point, and c) no points, we need to find the discriminant of the quadratic equation. The discriminant can help us determine the nature and number of solutions of a quadratic equation.
The discriminant (Δ) is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the standard form (ax^2 + bx + c = 0).
a) For two points of intersection:
Since the line intersects the quadratic at two points, it means the discriminant is greater than 0. Therefore, Δ > 0.
Substituting the values into the quadratic equation, we have kx^2 - 5x + 2 = -3x + 4.
Rearranging the equation, we get kx^2 - 2x - 2 = 0.
Comparing this equation to the standard form, we find a = k, b = -2, and c = -2.
Now, we calculate the discriminant:
Δ = b^2 - 4ac
Δ = (-2)^2 - 4(k)(-2)
Δ = 4 + 8k
For two points of intersection, Δ needs to be greater than 0:
4 + 8k > 0
8k > -4
k > -0.5
Therefore, any value of k greater than -0.5 will result in the line intersecting the quadratic at two points.
b) For one point of intersection:
When the line intersects the quadratic at one point, the discriminant is equal to 0. In this case, Δ = 0.
Using the same quadratic equation, we calculate the discriminant:
Δ = 4 + 8k
For one point of intersection, Δ needs to be equal to 0:
4 + 8k = 0
8k = -4
k = -0.5
Therefore, k = -0.5 is the value that will result in the line intersecting the quadratic at one point.
c) For no intersection:
If the line and the quadratic do not intersect, it means the discriminant is less than 0. Thus, Δ < 0.
Using the same quadratic equation, we find the discriminant:
Δ = 4 + 8k
For no intersection, Δ needs to be less than 0:
4 + 8k < 0
8k < -4
k < -0.5
Therefore, any value of k less than -0.5 will result in no intersection between the line and the quadratic equation.