X2-6x-11=0

To solve the quadratic equation \( x^2 - 6x - 11 = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this equation, the coefficients are: - \( a = 1 \) - \( b = -6 \) - \( c = -11 \) Now, let's plug these values into the quadratic formula. 1. Calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-11) = 36 + 44 = 80 \] 2. Now apply the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{80}}{2 \cdot 1} = \frac{6 \pm \sqrt{80}}{2} \] 3. Simplify \( \sqrt{80} \): \[ \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} \] 4. Substitute back into the formula: \[ x = \frac{6 \pm 4\sqrt{5}}{2} \] 5. Simplify further: \[ x = 3 \pm 2\sqrt{5} \] Thus, the solutions to the equation \( x^2 - 6x - 11 = 0 \) are: \[ x = 3 + 2\sqrt{5} \quad \text{and} \quad x = 3 - 2\sqrt{5} \]