Quadratic Equations

1. If the graph of y = 9(x-1)^2 - m - 6 intersects the x-axis, find the smallest value of m.
y = 9(x - 1)^2 - m - 6
y = 9(x^2 - 2x + 1) - m - 6
y = 9x^2 - 18x + 9 - m - 6
y = 9x^2 - 18x + 3 - m
Intersecting the x-axis there is real roots for y = 9x^2 - 18x + 3 - m = 0
Discriminant >= 0
18^2 - (4)(9)(3 - m) >= 0
324 - 36(3 - m) >= 0
9 >= 3 - m
m >= -6
The smallest value of m is -6
2. The expression 2x^2 - mx + 1 has a minimum value -7 for all values of x. Find the value of m.
2x^2 - mx + 1
= 2(x^2 - mx/2 + 1/2)
= 2[x^2 - mx/2 + (m/4)^2 +1/2 - (m/4)^2]
= 2(x - m/4)^2 + 1 - m^2/8
The minimum value happens when the braketed terms is zero, and the minimum value is 1 - m^2/8 = -7
8 = m^2/8
m^2 = 64
m = +/-8