Find the range of values of k for which the equation x2 + 2(k – 3)x + 33 – k = 0 has no roots.

Answer 1

a quadratic equation has no real roots if the discriminant, b^2 -4ac <0

a=1

b=2(k-3)

c= 33-k)

b^2 -4ac

=[2(k-3)]^2-4(33-k)

=4(k^2 -6k + 9) -33+k

=4k^2 -24k +36 -33 + k

=4k^2 -23k + 3

k= 23/8 + or - (1/8)sqr(23^2 -4(4)(3))

= (23 + or - sqr481)/8 = about 0.134, or 5.62

if 0.134 < k < 5.62, the discriminant <0 and there are no real roots

if k=0, the discriminant =+3

if k=1 the discriminant =-16

if k=2 the discriminant = -27

if k=3 the discriminant = -30

if k=4 the discriminant = -25

if k=5 the discriminant = -12

if k= 6 the discriminant = +9

in that range for k, 0.134 to 5.62 there are two imaginary roots