Find the set of values of k for which the line y=kx-4 intersects the curve y=x^2-2x at two distinct points.
Please help me solve this problem, 10 points?
Please help me solve this problem, 10 points?
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y=kx-4 and y=x^2-2x
intersects so kx-4=x^2-2x
simplify into the form: x^2 - (k+2)x + 4 = 0
when it comes to points of intersection, you look a the discriminant D=b^2-4ac, when ax^2+bx+c=0
intersects at 2 points so D > 0
(k+2)^2 - 4*1*4 > 0
k^2 + 4k -12 > 0
Change it to k^2 + 4k -12 = 0
and solve for k
k=2 or k=-6
Now you draw a graph (parabola shape of U on x-axis)
Looking back at k^2 + 4k -12 > 0, we want all the values that are above the x-axis (>0)
| |
\ / +
\ /
--------0------0---------->
-6 \ _ _ / 2 -
That occurs when k<-6 or k>2
Hope the diagram isn't too confusing :)
intersects so kx-4=x^2-2x
simplify into the form: x^2 - (k+2)x + 4 = 0
when it comes to points of intersection, you look a the discriminant D=b^2-4ac, when ax^2+bx+c=0
intersects at 2 points so D > 0
(k+2)^2 - 4*1*4 > 0
k^2 + 4k -12 > 0
Change it to k^2 + 4k -12 = 0
and solve for k
k=2 or k=-6
Now you draw a graph (parabola shape of U on x-axis)
Looking back at k^2 + 4k -12 > 0, we want all the values that are above the x-axis (>0)
| |
\ / +
\ /
--------0------0---------->
-6 \ _ _ / 2 -
That occurs when k<-6 or k>2
Hope the diagram isn't too confusing :)