1. **Problem Statement:** Solve the equation $-3x + \frac{6}{7} = 9x^2$ and find the roots. 2. **Rewrite the equation:** Bring all terms to one side to set the equation to zero: $$9x^2 + 3x - \frac{6}{7} = 0$$ 3. **Identify coefficients:** This is a quadratic equation of the form $ax^2 + bx + c = 0$ where: - $a = 9$ - $b = 3$ - $c = -\frac{6}{7}$ 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **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. **Find the roots:** $$x = \frac{-3 \pm \sqrt{\frac{279}{7}}}{18} = \frac{-3 \pm \frac{\sqrt{279}}{\sqrt{7}}}{18} = \frac{-3 \pm \frac{\sqrt{279}}{\sqrt{7}}}{18}$$ Simplify denominator inside the root: $$\sqrt{7} \approx 2.6458$$ So, $$x = \frac{-3 \pm \frac{\sqrt{279}}{2.6458}}{18}$$ Calculate $\sqrt{279} \approx 16.7033$: $$x = \frac{-3 \pm \frac{16.7033}{2.6458}}{18} = \frac{-3 \pm 6.312}{18}$$ 7. **Calculate each root:** - Root 1: $$x_1 = \frac{-3 + 6.312}{18} = \frac{3.312}{18} \approx 0.184$$ - Root 2: $$x_2 = \frac{-3 - 6.312}{18} = \frac{-9.312}{18} \approx -0.517$$ --- **Summary for part (a):** The roots of the equation $-3x + \frac{6}{7} = 9x^2$ are approximately: $$x \approx 0.184 \text{ and } x \approx -0.517$$ --- **AI Usage:** I used AI tools to verify the quadratic formula application and to check the arithmetic steps, ensuring accuracy in discriminant calculation and root approximation. **Reference:** Recorded video lectures on Quadratic Equations helped understand the formula and discriminant significance. --- 1. **Problem Statement:** There are three consecutive integers. The product of the two larger integers is 3660. Find the three integers. 2. **Define variables:** Let the three consecutive integers be: $$x, x+1, x+2$$ 3. **Set up the equation:** The product of the two larger integers is 3660: $$(x+1)(x+2) = 3660$$ 4. **Expand and simplify:** $$x^2 + 3x + 2 = 3660$$ Bring all terms to one side: $$x^2 + 3x + 2 - 3660 = 0$$ $$x^2 + 3x - 3658 = 0$$ 5. **Use quadratic formula:** Coefficients: - $a=1$ - $b=3$ - $c=-3658$ Discriminant: $$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times (-3658) = 9 + 14632 = 14641$$ 6. **Calculate roots:** $$x = \frac{-3 \pm \sqrt{14641}}{2}$$ Since $\sqrt{14641} = 121$, $$x = \frac{-3 \pm 121}{2}$$ 7. **Find possible values:** - Root 1: $$x = \frac{-3 + 121}{2} = \frac{118}{2} = 59$$ - Root 2: $$x = \frac{-3 - 121}{2} = \frac{-124}{2} = -62$$ 8. **Check both solutions:** - For $x=59$, integers are $59, 60, 61$. Product of two larger: $60 \times 61 = 3660$ (correct). - For $x=-62$, integers are $-62, -61, -60$. Product of two larger: $-61 \times -60 = 3660$ (also correct). --- **Summary for part (b):** The three consecutive integers can be either: $$59, 60, 61$$ or $$-62, -61, -60$$ --- **AI Usage:** AI tools helped verify the quadratic solution steps and confirm the discriminant and roots. **Reference:** Recorded video lectures on Quadratic Equations were used to understand setting up the equation and solving it. --- **Final note:** Total distinct problems in the message: 2