The product of the roots is given by c/a = 24/1 = 24.

The product of the roots is given by c/a = 15/1 = 15.

The sum of the roots is given by -b/a = 4/2 = 2.

The sum of the roots is given by -b/a = -12/3 = -4.

The product of the roots is k = 2 * 4 = 8.

For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, thus k < -8 or k > 8.

For no real roots, the discriminant must be less than zero: (-4)² - 4*1*k < 0, hence k > 4.

Since the roots are equal, k = 4 + 4 = 8.

The discriminant D = b² - 4ac = 6² - 4*3*2 = 36 - 24 = 12.

The discriminant D = b² - 4ac = (-12)² - 4*3*12 = 144 - 144 = 0.

The discriminant is b² - 4ac = (-2)² - 4*1*1 = 0.

The discriminant is b² - 4ac = (-4)² - 4*1*4 = 0.

The vertex can be found using the formula x = -b/2a. Here, x = 2, and substituting back gives y = -1.

The vertex can be found using the formula x = -b/2a = 6/2 = 3. Substituting x = 3 into the equation gives y = -1.

The equation x² + 4x + 4 = 0 has roots -2 and -2, which are both negative.