Quadratic Equation 3F7B76

1. **State the problem:** Solve the quadratic equation $$12 - 8w + w^2 = 0$$. 2. **Rewrite the equation in standard form:** The standard form of a quadratic equation is $$aw^2 + bw + c = 0$$. Here, rewrite as $$w^2 - 8w + 12 = 0$$. 3. **Identify coefficients:** $$a = 1$$, $$b = -8$$, $$c = 12$$. 4. **Use the quadratic formula:** $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-8)^2 - 4 \times 1 \times 12 = 64 - 48 = 16$$. 6. **Calculate the roots:** $$w = \frac{-(-8) \pm \sqrt{16}}{2 \times 1} = \frac{8 \pm 4}{2}$$. 7. **Find each solution:** - $$w_1 = \frac{8 + 4}{2} = \frac{12}{2} = 6$$ - $$w_2 = \frac{8 - 4}{2} = \frac{4}{2} = 2$$ 8. **Final answer:** The solutions to the equation are $$w = 6$$ and $$w = 2$$.