Questions: Solve the equation by the method of your choice. (3x+1)(x+2)=1 The solution set is . (Simplify your answer. Type an exact answer, using radicals and i in the expression. Use a comma to separate answers as needed.)

Solution Steps

To solve the equation \((3x + 1)(x + 2) = 1\), we can follow these steps:

1. Expand the left-hand side of the equation.
2. Move all terms to one side to set the equation to zero.
3. Solve the resulting quadratic equation using the quadratic formula.

We start with the equation \((3x + 1)(x + 2) = 1\). Expanding the left-hand side gives us: \[ 3x^2 + 6x + x + 2 = 1 \] which simplifies to: \[ 3x^2 + 7x + 2 = 1 \]

Next, we move all terms to one side of the equation: \[ 3x^2 + 7x + 2 - 1 = 0 \] This simplifies to: \[ 3x^2 + 7x + 1 = 0 \]

We apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = 7\), and \(c = 1\): \[ x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \] Calculating the discriminant: \[ 7^2 - 4 \cdot 3 \cdot 1 = 49 - 12 = 37 \] Thus, the solutions are: \[ x = \frac{-7 \pm \sqrt{37}}{6} \] This results in two solutions: \[ x_1 = \frac{-7 - \sqrt{37}}{6}, \quad x_2 = \frac{-7 + \sqrt{37}}{6} \]

Final Answer