Solve the equation : log 3 (x^2 + 4x + 12) =2

First, we'll verify if the argument of the logarithm is
positive. For this reason, we'll calculate the discriminant of the
quadratic.

If the discriminant is negative and the
coefficient of x^2 is positive, then the expression x^2 + 4x + 12 is positive for any
value of x.

delta = b^2 -
4ac

We'll identify the coefficients
a,b,c:

a = 1

b =
4

c = 12

delta = 16 -
4*12

delta = 16 - 48

delta =
-32

Since delta is negative and a is positive, the
expression x^2 + 4x + 12 > 0.

Now, we'll solve the
equation. We'll take anti-logarithm:

x^2 + 4x + 12 =
3^2

x^2 + 4x + 12 = 9

We'll
subtract 9 both sides:

x^2 + 4x + 12 - 9 =
0

We'll combine like
terms:

x^2 + 4x + 3 = 0

We'll
apply the quadratic formula:

x1 = [-4 +/-
sqrt(16-12)]/2

x1 =
(-4+2)/2

x1 = -1

x2 =
-3

Since all values of x are admissible, we'll not reject
either of resulted roots.

The solutions of
the equation are: {-3 ; -1}.