\( x = \frac { - 6 \pm \sqrt { 6 ^ { 2 } - 4 ( 1 ) } } { 2 ( 1 ) } \)

Certainly! Let's simplify the given quadratic formula step by step. ### Given Expression: \[ x = \frac{ -6 \pm \sqrt{6^2 - 4 \cdot 1 }}{2 \cdot 1} \] ### Step 1: Simplify Inside the Square Root Calculate the discriminant (\( \Delta \)): \[ \Delta = b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot 1 = 36 - 4 = 32 \] ### Step 2: Substitute Back into the Formula \[ x = \frac{ -6 \pm \sqrt{32} }{2} \] ### Step 3: Simplify the Square Root \[ \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \] ### Step 4: Substitute and Simplify the Expression \[ x = \frac{ -6 \pm 4\sqrt{2} }{2} = \frac{ -6 }{2} \pm \frac{4\sqrt{2}}{2} = -3 \pm 2\sqrt{2} \] ### Final Solutions: \[ x = -3 + 2\sqrt{2} \quad \text{and} \quad x = -3 - 2\sqrt{2} \] ### Approximate Numerical Values: If you'd like approximate decimal values: \[ \sqrt{2} \approx 1.4142 \] \[ x \approx -3 + 2(1.4142) = -3 + 2.8284 = -0.1716 \] \[ x \approx -3 - 2.8284 = -5.8284 \] ### Conclusion: The solutions to the quadratic equation are: \[ x = -3 + 2\sqrt{2} \quad \text{and} \quad x = -3 - 2\sqrt{2} \] Or, approximately: \[ x \approx -0.1716 \quad \text{and} \quad x \approx -5.8284 \]