(1) \( x^{2}-3 x-2=0 \)

To solve the quadratic equation: \[ x^{2} - 3x - 2 = 0 \] we can use the **quadratic formula**, which is applicable to any quadratic equation of the form \( ax^{2} + bx + c = 0 \). The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] ### Step-by-Step Solution 1. **Identify the coefficients** in the equation \( x^{2} - 3x - 2 = 0 \): - \( a = 1 \) (coefficient of \( x^{2} \)) - \( b = -3 \) (coefficient of \( x \)) - \( c = -2 \) (constant term) 2. **Plug the coefficients** into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{(-3)^{2} - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \] 3. **Simplify inside the square root**: \[ (-3)^{2} = 9 \\ 4 \cdot 1 \cdot (-2) = -8 \\ \] So, \[ x = \frac{3 \pm \sqrt{9 - (-8)}}{2} = \frac{3 \pm \sqrt{17}}{2} \] 4. **Express the final solutions**: \[ x = \frac{3 + \sqrt{17}}{2} \quad \text{or} \quad x = \frac{3 - \sqrt{17}}{2} \] ### Approximate Numerical Values If you need approximate decimal values: \[ \sqrt{17} \approx 4.1231 \] So, \[ x \approx \frac{3 + 4.1231}{2} = \frac{7.1231}{2} \approx 3.5616 \] \[ x \approx \frac{3 - 4.1231}{2} = \frac{-1.1231}{2} \approx -0.5616 \] ### Summary of Solutions - **Exact Solutions**: \[ x = \frac{3 + \sqrt{17}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{17}}{2} \] - **Approximate Decimal Solutions**: \[ x \approx 3.5616 \quad \text{and} \quad x \approx -0.5616 \] These are the two solutions to the quadratic equation \( x^{2} - 3x - 2 = 0 \).