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

Question

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

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

c) y = 2x^2 + 3x + 1

Answer 1

a) y = -5(x + 1)^2 - 3
since the vertex is (-1, -3) , which is already below the x-axis,
and the parabola opens down, there can't be any real roots

b) y = 2(x-1)(x+3)
since it is factored, the roots are 1 and -3, that is, there are two roots

c) y = 2x^2 + 3x + 1
the discriminant of the corresponding quadratic equation
= b^2 - 4ac
= 9 - 4(2)(3) = negative, so there can't be any real roots