Solve the equation : `log_3 (x^2 + 4x + 12) =2`

sciencesolve | Certified Educator

You need to use the property of logarithmic function, such that:

`log_3 (x^2 + 4x + 12) = 2 => log_3 (x^2 + 4x + 12) = 2log_3 3`

`log_3 (x^2 + 4x + 12) = log_3 3^2 => log_3 (x^2 + 4x + 12)

= `log_3 9 =gt x^2 + 4x + 12 = 9 =gt x^2 + 4x = -3` `

You need to complete the square `x^2 + 4x` using the formula `(a + b)^2 = a^2 + 2ab + b^2` , such that:

`x^2 + 4x + 4 = 4 - 3 => (x + 2)^2 = 1 => x + 2 = +-1`

`x_1 = -2 + 1 => x_1 = -1`

`x_2 = -2 - 1 => x_2 = -3`

You need to test the values `x = -1` and `x = -3` in logarithmic equation such that:

`log_3 ((-1)^2 + 4(-1) + 12) = 2 => log_3 (1 - 4 + 12) = 2

`=>log_3 9 = 2 =gt log_3 3^2 = 2 =gt 2 log_3 3 = 2 =gt 2 = 2`

`log_3 ((-3)^2 + 4(-3) + 12) = 2 => log_3 (9 -12 + 12) = 2

=> `log_3 9 = 2 =gt 2 = 2` `

Hence, testing both values `x = -1` and `x = -3` yields that the logarithmic equation holds.

giorgiana1976 | Student

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

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}.