Question 88, 1.7.34 Part 1 of 2 \( \begin{array}{l}\text { HW Score: } 89.72 \%, 84.33 \\ \text { of } 94 \text { points } \\ \text { Use the quadratic formula to solve the following equation. If the solutions involve square roots, give both } \\ \text { the exact and approximate solutions. } \\ 3 x^{2}=3 x-2\end{array} \)

To solve the quadratic equation using the quadratic formula, follow these steps: ### Given Equation: \[ 3x^{2} = 3x - 2 \] ### Step 1: Rewrite the Equation in Standard Form Bring all terms to one side to set the equation to zero: \[ 3x^{2} - 3x + 2 = 0 \] ### Step 2: Identify the Coefficients The standard quadratic form is: \[ ax^{2} + bx + c = 0 \] Here: - \( a = 3 \) - \( b = -3 \) - \( c = 2 \) ### Step 3: Apply the Quadratic Formula The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Plugging in the values: \[ x = \frac{-(-3) \pm \sqrt{(-3)^{2} - 4 \cdot 3 \cdot 2}}{2 \cdot 3} \] \[ x = \frac{3 \pm \sqrt{9 - 24}}{6} \] \[ x = \frac{3 \pm \sqrt{-15}}{6} \] ### Step 4: Simplify the Square Root Since the discriminant (\( b^{2} - 4ac = -15 \)) is negative, the solutions involve imaginary numbers. \[ \sqrt{-15} = i\sqrt{15} \] ### Step 5: Write the Exact Solutions \[ x = \frac{3 \pm i\sqrt{15}}{6} \] This can be simplified further by dividing numerator and denominator by 3: \[ x = \frac{1 \pm i\sqrt{15}}{2} \] ### Step 6: Provide Approximate Solutions Calculate the numerical value of the imaginary part: \[ \sqrt{15} \approx 3.87298 \] \[ \frac{\sqrt{15}}{6} \approx 0.6455 \] So, the approximate solutions are: \[ x \approx \frac{1}{2} \pm 0.6455i \] \[ x \approx 0.5 \pm 0.6455i \] ### **Final Answers:** - **Exact Solutions:** \[ x = \frac{3 + i\sqrt{15}}{6} \quad \text{and} \quad x = \frac{3 - i\sqrt{15}}{6} \] *Alternatively simplified:* \[ x = \frac{1}{2} + \frac{i\sqrt{15}}{6} \quad \text{and} \quad x = \frac{1}{2} - \frac{i\sqrt{15}}{6} \] - **Approximate Solutions:** \[ x \approx 0.5 + 0.645i \quad \text{and} \quad x \approx 0.5 - 0.645i \] --- **Note:** Since the discriminant is negative, the solutions are complex (imaginary) numbers.