the equation kx^2+4kx+3=0 where k is a constant and has no real roots. prove that 0≤k<3/4

Answer 1

(4k)² - 4(k)(3) < 0

16k² - 12k < 0

2k(8k - 6) < 0

Either

k > 0 and 8k - 6 < 0 implying k < 3/4

Or

k < 0 and 8k - 6 > 0 implying k > 3/4, not possible since k < 0.

Thus 0 < k < 3/4 results in non-real roots.