Questions from http://www.sgforums.com/forums/2297/topics/311161
1) Show that x2 + hx + k= 0 has two distinct roots for all negative values of k.
2) Find the values of k for which the x axis is a tangent to the curve y = 3x2 - 8x+ 5 - k. For each value of k, find the coordinates for the point of tangency.
3) Find the range of values of k for which the equation x2 + 5x + 3 = k has two real and distinct roots.
Answer:
1)
For
x2 + hx + k = 0
discriminant = h2 - 4(1)(k) = h2 -4k
h2 ≥ 0
also, for negative values of k, -4k > 0
Therefore h2 - 4k is always +ve, ie >0
Thus, since discriminant is always > 0, equation will always have 2 distinct roots.
2) x-axis tangent to the curve => curve touches the line y=0 only 1 time
so 3x2 - 8x + 5 - k = 0 has only 1 root (real and equal roots).
Discriminant = 0 for real and equal roots
64 - 4(3)(5 - k) = 0
4 + 12k = 0
k = -⅓
Find value of x: 2 methods
1st method:
3x2 - 8x + 16/3 = 0
9x2 - 24x + 16 = 0
(3x - 4)2 = 0
x = 4/3
and y = 0 (since it is on x-axis)
Thus
coordinates of tangential point: (4/3, 0)
2nd method:
Obviously, for an x2 curve, ie curve with smiling face, the minimum point in this case will be the point at tangent with the x-axis.
so dy/dx = 6x-8
x = 4/3
Similarly, y = 0 (since it is on x-axis)
Thus
coordinates of tangential point: (4/3, 0)
