Determine for what values of c, f(x)=4x^2-(2c+8)x+4 has no real roots.

For f(x) not to have any real roots, its discriminant, b²-4ac < 0. Plugging in the coefficients from f(x), this means (-(2c+8))²-4x4x4 < 0.This means (-(2c+8))² < 4x4x4So 4c²+32c+64 < 64 by multiplying out the brackets.This means 4c²+32c < 0So c²+8c < 0, which factorised gives c(c+8) < 0As this has roots at c=-8 and c=0, by plotting a graph we can see that the range of values of -8<x<0.