Questions: Solve the equation by the method of your choice. (3 x+3)(x+2)=2 The solution set is .

Solution Steps

To solve the equation \((3x + 3)(x + 2) = 2\), we can first expand the left-hand side to form a quadratic equation. Then, we can rearrange the equation to set it to zero and use the quadratic formula to find the values of \(x\).

We start with the equation: \[ (3x + 3)(x + 2) = 2 \] Expanding the left-hand side gives: \[ 3x^2 + 6x + 3x + 6 = 2 \] which simplifies to: \[ 3x^2 + 9x + 6 = 2 \]

Next, we rearrange the equation to set it to zero: \[ 3x^2 + 9x + 6 - 2 = 0 \] This simplifies to: \[ 3x^2 + 9x + 4 = 0 \]

We apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = 9\), and \(c = 4\): \[ x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 3 \cdot 4}}{2 \cdot 3} \] Calculating the discriminant: \[ 9^2 - 4 \cdot 3 \cdot 4 = 81 - 48 = 33 \] Thus, the solutions are: \[ x = \frac{-9 \pm \sqrt{33}}{6} \] This results in two solutions: \[ x_1 = \frac{-9 - \sqrt{33}}{6}, \quad x_2 = \frac{-9 + \sqrt{33}}{6} \]

Final Answer