1. **Problem a:** Solve the quadratic equation $$-3x + \frac{6}{7} = 9x^2$$ and find its roots. 2. **Step 1:** Rearrange the equation to standard quadratic form $$ax^2 + bx + c = 0$$. $$9x^2 + 3x - \frac{6}{7} = 0$$ 3. **Step 2:** Identify coefficients: $$a = 9, b = 3, c = -\frac{6}{7}$$. 4. **Step 3:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Step 4:** Calculate the discriminant: $$\Delta = b^2 - 4ac = 3^2 - 4 \times 9 \times \left(-\frac{6}{7}\right) = 9 + \frac{216}{7} = \frac{63}{7} + \frac{216}{7} = \frac{279}{7}$$ 6. **Step 5:** Calculate roots: $$x = \frac{-3 \pm \sqrt{\frac{279}{7}}}{18} = \frac{-3 \pm \frac{\sqrt{279}}{\sqrt{7}}}{18} = \frac{-3 \pm \frac{\sqrt{279}}{\sqrt{7}}}{18}$$ 7. **Step 6:** Simplify the roots: $$x = \frac{-3}{18} \pm \frac{\sqrt{279}}{18\sqrt{7}} = -\frac{1}{6} \pm \frac{\sqrt{279}}{18\sqrt{7}}$$ --- 8. **Problem b:** There are three consecutive integers. The product of the two larger integers is 3660. Find the three integers. 9. **Step 1:** Let the three consecutive integers be $$x$$, $$x+1$$, and $$x+2$$. 10. **Step 2:** The product of the two larger integers is: $$(x+1)(x+2) = 3660$$ 11. **Step 3:** Expand and form quadratic equation: $$x^2 + 3x + 2 = 3660$$ $$x^2 + 3x + 2 - 3660 = 0$$ $$x^2 + 3x - 3658 = 0$$ 12. **Step 4:** Use quadratic formula with $$a=1$$, $$b=3$$, $$c=-3658$$: $$x = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-3658)}}{2} = \frac{-3 \pm \sqrt{9 + 14632}}{2} = \frac{-3 \pm \sqrt{14641}}{2}$$ 13. **Step 5:** Calculate $$\sqrt{14641} = 121$$. 14. **Step 6:** Find roots: $$x = \frac{-3 \pm 121}{2}$$ - Positive root: $$x = \frac{-3 + 121}{2} = \frac{118}{2} = 59$$ - Negative root: $$x = \frac{-3 - 121}{2} = \frac{-124}{2} = -62$$ 15. **Step 7:** The three consecutive integers are either $$59, 60, 61$$ or $$-62, -61, -60$$. 16. **Step 8:** Verify product: $$60 \times 61 = 3660$$ and $$-61 \times -60 = 3660$$ both true. --- **Use of AI:** - I used AI tools like ChatGPT to verify the quadratic formula steps and to confirm the factorization and root calculations. - AI helped me understand how to set up the equations from word problems and check arithmetic. - I referred to recorded video lectures on Quadratic Equations to ensure correct application of formulas and problem-solving strategies. **Final answers:** - a) Roots are $$x = -\frac{1}{6} \pm \frac{\sqrt{279}}{18\sqrt{7}}$$. - b) The three consecutive integers are either $$59, 60, 61$$ or $$-62, -61, -60$$.