Firstly, we need to rearrange the equation 2x^2 + 14x + 24 = 0 to its standard form, which is already given.
To find the roots of this quadratic equation, we utilize the Quadratic Formula:
x = [-b ± sqrt(b^2 - 4ac)] / 2a
Our coefficients for the quadratic equation are a = 2, b = 14, and c = 24.
Before we can find the roots, we need to check the discriminant (b^2 - 4ac). The discriminant will help us to know the number of roots in the given quadratic equation.
So our discriminant will be:
(14^2) - (4*2*24) = 196 - 192 = 4
Since the discriminant is greater than zero, it means that we are dealing with two distinct roots.
Now let's substitute a, b, and c into the quadratic formula to find our roots.
For the root1 which is x:
x = [-14 + sqrt(4)] / 2*2
x = [-14 + 2] / 4
x = -12 / 4
x = -3.0
For the root2 which is y:
y = [-14 - sqrt(4)] / 2*2
y = [-14 - 2] / 4
y = -16 / 4
y = -4.0
So, the solutions to the quadratic equation 2x^2 + 14x + 24 = 0 are x=-3 and y=-4.