Let's start by solving the given equation step-by-step. The aim is to solve the quadratic equation [tex]\(3x^2 + 12x = 9\)[/tex] by completing the square.
### Step 1: Rewrite the Equation
First, we need to move all terms to one side of the equation to set it equal to zero:
[tex]\[ 3x^2 + 12x - 9 = 0 \][/tex]
### Step 2: Simplify the Equation
Next, we divide the entire equation by 3 in order to simplify it:
[tex]\[ x^2 + 4x - 3 = 0 \][/tex]
### Step 3: Rearrange the Equation
Now, we'll move the constant term to the right side of the equation:
[tex]\[ x^2 + 4x = 3 \][/tex]
### Step 4: Complete the Square
To complete the square, we need to add and subtract a specific value on the left side. The value to add and subtract is determined by taking half of the coefficient of [tex]\(x\)[/tex], squaring it, and then balancing the equation:
[tex]\[
\left(\frac{4}{2}\right)^2 = 4
\][/tex]
Add and subtract 4 on the left side:
[tex]\[ x^2 + 4x + 4 - 4 = 3 \][/tex]
[tex]\[ (x + 2)^2 - 4 = 3 \][/tex]
### Step 5: Simplify the Equation
Now we move the -4 to the right side of the equation:
[tex]\[ (x + 2)^2 = 3 + 4 \][/tex]
[tex]\[ (x + 2)^2 = 7 \][/tex]
### Step 6: Solve for [tex]\(x\)[/tex]
To solve for [tex]\(x\)[/tex], take the square root of both sides:
[tex]\[ x + 2 = \pm \sqrt{7} \][/tex]
Subtract 2 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = -2 \pm \sqrt{7} \][/tex]
### Conclusion
Thus, the solutions to the quadratic equation [tex]\(3x^2 + 12x = 9\)[/tex] when solved by completing the square are:
[tex]\[ x = -2 \pm \sqrt{7} \][/tex]
So, the correct answer is:
[tex]\[
\boxed{\text{B. } x = -2 \pm \sqrt{7}}
\][/tex]
These solutions evaluate roughly to:
[tex]\[ \left[-4.645751311064591, 0.6457513110645907\right] \][/tex]
These match the numerical solutions, confirming our step-by-step solution is correct.
