Certainly! Let's analyze and solve the given equation step-by-step:
We need to solve the equation:
[tex]\[ 3x^2 - 6x - 4 = -\frac{2}{x + 3} + 1 \][/tex]
### Step 1: Identify the type of functions
- The left-hand side (LHS) of the equation is a quadratic function: [tex]\( 3x^2 - 6x - 4 \)[/tex].
- The right-hand side (RHS) of the equation is a rational function: [tex]\( -\frac{2}{x+3} + 1 \)[/tex].
### Step 2: Set the functions equal to each other
We'll set the LHS equal to the RHS to find the values of [tex]\( x \)[/tex]:
[tex]\[ 3x^2 - 6x - 4 = -\frac{2}{x + 3} + 1 \][/tex]
### Step 3: Solve the equation for x
To find the solutions, we will find the points of intersection between the quadratic function and the rational function. The solutions to this equation are the [tex]\( x \)[/tex]-values where the two curves intersect on a graph.
### Step 4: Determine the approximate solutions from the results
The solutions obtained from this equation are:
- [tex]\(-0.54752462742284 + 0.e-22I\)[/tex]
- [tex]\(-3.0485756474747 - 0.e-22I\)[/tex]
- [tex]\(2.59610027489754 - 0.e-20*I\)[/tex]
The first two solutions are complex numbers with significant imaginary components close to zero, which implies they are not real numbers. However, the third solution, [tex]\( 2.59610027489754 \)[/tex], is a real number close to 2.60.
### Step 5: Selecting the correct answer
From the provided choices:
A. [tex]\( x \approx 2.60 \)[/tex]
B. [tex]\( x \approx 0.18 \)[/tex]
C. [tex]\( x \approx 0.64 \)[/tex]
D. [tex]\( x \approx 0.33 \)[/tex]
The approximate solution to the equation, which is the real solution close to one of these values, is:
[tex]\[ x \approx 2.60 \][/tex]
Thus, the correct answer is:
A. [tex]\( x \approx 2.60 \)[/tex]
