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)
To determine the value of k that results in the intersection of the line y = -3x + 4 with the quadratic equation y = kx^2 - 5x + 2, we need to find the discriminant (D) of the quadratic equation.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. The value of the discriminant tells us how many solutions (or points of intersection) the quadratic equation has with another equation, such as a line.
Now, let's solve for k in each case:
a) Two Points of Intersection:
For two points of intersection, the discriminant (D) must be greater than zero.
D = (-5)^2 - 4(1)(2) = 25 - 8 = 17.
Since D > 0, there will be two points of intersection between the line y = -3x + 4 and the quadratic equation.
b) One Point of Intersection:
For one point of intersection, the discriminant (D) must be equal to zero.
D = (-5)^2 - 4(1)(2) = 25 - 8 = 17.
Since D = 0, there will be one point of intersection between the line y = -3x + 4 and the quadratic equation.
c) No Points of Intersection:
For no points of intersection, the discriminant (D) must be less than zero.
D = (-5)^2 - 4(1)(2) = 25 - 8 = 17.
Since D < 0, there will be no point of intersection between the line y = -3x + 4 and the quadratic equation.
In summary:
a) For two points of intersection, the value of k can be any real number.
b) For one point of intersection, the value of k will remain the same, regardless of the line equation.
c) For no points of intersection, the value of k can be any real number except for a value that results in a positive discriminant (D > 0).