To solve the quadratic equation 2x² + 12x + 14 = 0, we can use the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / 2a
In our case, the coefficients are:
- a = 2
- b = 12
- c = 14
First, we need to calculate the discriminant (b² – 4ac):
b² = 12² = 144
4ac = 4 * 2 * 14 = 112
Now, we can find the discriminant:
Discriminant = 144 – 112 = 32
Since the discriminant is positive, we will have two real and distinct solutions. Now plug the values into the quadratic formula:
x = (-12 ± √32) / (2 * 2)
Next, we compute the square root of 32:
√32 = 4√2 (because 32 can be simplified to 16 * 2)
Now substitute this back into our equation:
x = (-12 ± 4√2) / 4
This can be simplified to:
x = -3 ± √2
So, the final solutions are:
- x = -3 + √2
- x = -3 – √2
These are the two solutions for the equation 2x² + 12x + 14 = 0.