Determine the number of roots, using the most efficient method possible.

a) y = - 5 (x + 1) 2 - 3

b) y = 2 (x - 1) (x + 3)

c) y = 2x2 + 3x + 1

2 roots in every case, not all are real.

Step-by-step explanation:

All of these equations are quadratic equations (degree 2). Every quadratic has two roots. They may be identical (looks like 1 root), and they may be complex (zero real roots), but there are always 2 of them.

a) the y-value of the vertex is negative and the parabola opens downward (leading coefficient - 5), so there are no real zeros and both roots are complex.

b) each binomial factor contributes a root. Both roots are real.

c) the discriminant is positive, (3²-4·2·1=1), so both roots are real.