F.4 Quadratic equations...

(1)In a rectangular coordinateplane， if the length of a line segment joining the two points
(4，7) and (k， k+4) is 5， find the values of k.
Sol
25=(k－4)^2+(k+4－7)^2
25=k^2－8k+16+k^2－6k+9
2k^2－14k=0
k^2－7k=0
k(k－7)=0
k=0 or k=7

(2)If the following quadratic equation has real roots and k is a constant， find therange
of k，k(2x+1) = 2－k(x^2)
Sol
k(2x+1)=2－kx^2
kx^2+2kx+k－2=0
kx^2+2kx+(k－2)=0
D=(2k)^2－4*k*(k－2)>=0
k^2－k(k－2)>=0
k^2－k^2+2k>=0
2k>=0
k>=0

1. √[(k-4)^2+(k+4-7)^2] = 5
k^2-8k+16+k^2-6k+9 = 25
2k^2-14k+25=25
k^2-7k=0
k=0 or k =7

2. k(2x+1) = 2-kx^2
2kx+k = 2-kx^2
kx^2+2kx+(k-2) = 0

∵has real roots
∴ ∆≥0
(2k)^2-4(k)(k-2) ≥ 0
4k^2-4k^2+8k ≥ 0
8k ≥ 0
k≥0