To find the zeros of the quadratic function f(x) = 6x² + 12x + 7, we need to solve the equation f(x) = 0. This can be done using the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / 2a
In our case, the coefficients are:
- a = 6
- b = 12
- c = 7
First, we will calculate the discriminant (b² – 4ac):
Discriminant = b² – 4ac = 12² – 4(6)(7) = 144 – 168 = -24
Since the discriminant is negative, this means that the quadratic function does not cross the x-axis, and therefore, there are no real zeros; instead, there are two complex zeros.
Now, we can use the quadratic formula to find these complex zeros:
x = (-12 ± √(-24)) / (2 * 6)
√(-24) can be simplified as √(24) * i = 2√6 * i. So we now have:
x = (-12 ± 2√6i) / 12
This simplifies to:
x = -1 ± (√6/6)i
Therefore, the zeros of the function f(x) = 6x² + 12x + 7 are:
x = -1 + (√6/6)i and x = -1 – (√6/6)i.