Quadratic Sum Product 348277

1. **Stating the problem:** Solve the quadratic equation $$x^{2}-6x-6=0$$ using the sum-product method. 2. **Sum-product method formula:** For a quadratic equation $$x^{2}+bx+c=0$$, find two numbers whose sum is $$-b$$ and product is $$c$$. 3. **Identify coefficients:** Here, $$b=-6$$ and $$c=-6$$. 4. **Sum and product:** We need two numbers $$m$$ and $$n$$ such that: $$m+n = 6$$ (since sum is $$-b$$ and $$b=-6$$) $$m \times n = -6$$ 5. **Check possible pairs:** Factors of $$-6$$ are: - $$1$$ and $$-6$$ (sum = $$-5$$) - $$-1$$ and $$6$$ (sum = $$5$$) - $$2$$ and $$-3$$ (sum = $$-1$$) - $$-2$$ and $$3$$ (sum = $$1$$) None of these pairs sum to $$6$$. 6. **Conclusion:** Since no integer pair satisfies the sum-product condition, the sum-product method with integers does not work here. 7. **Alternative:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ where $$a=1$$, $$b=-6$$, $$c=-6$$. 8. **Calculate discriminant:** $$\Delta = (-6)^{2} - 4 \times 1 \times (-6) = 36 + 24 = 60$$ 9. **Calculate roots:** $$x = \frac{6 \pm \sqrt{60}}{2} = \frac{6 \pm 2\sqrt{15}}{2} = 3 \pm \sqrt{15}$$ **Final answer:** $$x = 3 + \sqrt{15}$$ or $$x = 3 - \sqrt{15}$$