Let's solve the given equation step-by-step:
[tex]\[\sqrt{3 - 3x} = 3 + \sqrt{3x + 2}\][/tex]
### Step 1: Isolate one of the square root terms
To start, we'll isolate one of the square root expressions. Let's isolate [tex]\(\sqrt{3 - 3x}\)[/tex].
[tex]\[\sqrt{3 - 3x} - \sqrt{3x + 2} = 3\][/tex]
### Step 2: Square both sides to eliminate the square roots
Next, we square both sides of the equation to remove the square roots:
[tex]\[(\sqrt{3 - 3x} - \sqrt{3x + 2})^2 = 3^2\][/tex]
Expanding the left-hand side using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[(3 - 3x) - 2 \sqrt{(3 - 3x)(3x + 2)} + (3x + 2) = 9\][/tex]
Simplify:
[tex]\[(3 - 3x) + (3x + 2) - 2 \sqrt{(3 - 3x)(3x + 2)} = 9\][/tex]
[tex]\[5 - 2 \sqrt{(3 - 3x)(3x + 2)} = 9\][/tex]
### Step 3: Isolate the square root term again
We now isolate the square root term:
[tex]\[-2 \sqrt{(3 - 3x)(3x + 2)} = 9 - 5\][/tex]
[tex]\[-2 \sqrt{(3 - 3x)(3x + 2)} = 4\][/tex]
Divide both sides by -2 to get:
[tex]\[\sqrt{(3 - 3x)(3x + 2)} = -2\][/tex]
### Step 4: Square both sides again
Now, square both sides to eliminate the square root:
[tex]\[(3 - 3x)(3x + 2) = (-2)^2\][/tex]
[tex]\[ (3 - 3x)(3x + 2) = 4\][/tex]
### Step 5: Expand and simplify
Expand the left side:
[tex]\[ (3 - 3x)(3x + 2) = 9x + 6 - 9x^2 - 6x\][/tex]
[tex]\[ = 9 - 9x^2\][/tex]
So, we have:
[tex]\[9 - 9x^2 = 4\][/tex]
Subtract 4 from both sides to formulate a standard quadratic equation:
[tex]\[9 - 9x^2 - 4 = 0\][/tex]
[tex]\[9 - 9x^2 = 4\][/tex]
### Step 6: Solve the quadratic equation
[tex]\[-9x^2 + 5 = 0\][/tex]
Divide by -1:
[tex]\[9x^2 = 5\][/tex]
Divide by 9:
[tex]\[x^2 = \frac{5}{9}\][/tex]
Take the square root of both sides:
[tex]\[x = \pm \sqrt{\frac{5}{9}}\][/tex]
[tex]\[x = \pm \frac{\sqrt{5}}{3}\][/tex]
### Step 7: Verify potential solutions
We have two potential solutions: [tex]\(x = \frac{\sqrt{5}}{3}\)[/tex] and [tex]\(x = -\frac{\sqrt{5}}{3}\)[/tex]. However, we must check these solutions in the original equation to see if they create true statements or extraneous solutions.
Plugging [tex]\(x = \frac{\sqrt{5}}{3}\)[/tex] into the original equation will reveal whether it holds true.
### Step 8: Conclusion
Upon examination, neither of the potential solutions holds true. Therefore, the original equation has no valid solutions.
The equation:
[tex]\[\sqrt{3 - 3x} = 3 + \sqrt{3x + 2}\][/tex]
has no solution.
