Directions: Solve the quadratic and answer the following questions. A : $(x-2)^2 = (-3x^2) + 2^2 + 25x - P$ B: $\left(10y^2 - 3^2y + \frac{2}{3}\right) \times 3 + 10y = 0$ One root of equation A is $5$. Which of the following are the smallest root of equation A and largest root of equation B-respectively?

### Concept & Factorization We need to determine the roots for both Equation A and Equation B. After finding all roots, we simply compare them to select the smallest root from A and the largest root from B. $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ ### Step-by-Step Solution 1. **Find roots of Equation A:** As derived previously, Equation A simplifies to $4x^2 - 29x + 45 = 0$. The roots are $x = 5$ and $x = \frac{9}{4}$ (or $2.25$). The smallest root of Equation A is $\frac{9}{4}$. 2. **Simplify Equation B:** $\left(10y^2 - 3^2y + \frac{2}{3}\right) \times 3 + 10y = 0$ Expand the equation by multiplying by $3$: $30y^2 - 27y + 2 + 10y = 0$ $30y^2 - 17y + 2 = 0$ 3. **Find roots of Equation B:** Factorize by splitting the middle term. We need two numbers that multiply to $(30 \times 2 = 60)$ and add to $-17$. These are $-12$ and $-5$. $30y^2 - 12y - 5y + 2 = 0$ $6y(5y - 2) - 1(5y - 2) = 0$ $(6y - 1)(5y - 2) = 0$ The roots are $y = \frac{1}{6}$ and $y = \frac{2}{5}$. 4. **Identify the largest root of Equation B:** Compare $\frac{1}{6}$ (approx $0.166$) and $\frac{2}{5}$ (which is $0.4$). The largest root of Equation B is $\frac{2}{5}$. 5. **Combine the results:** Smallest root of A = $\frac{9}{4}$. Largest root of B = $\frac{2}{5}$. Respectively, the values are $\frac{9}{4}, \frac{2}{5}$. ### Exam Strategy & Shortcut For Equation B, once you reach $30y^2 - 17y + 2 = 0$, you can find the roots using the shortcut of changing the signs of the split middle terms and dividing by $a$: The split terms are $-12, -5$. Change signs: $12, 5$. Divide by $30$: $\frac{12}{30}, \frac{5}{30} \implies \frac{2}{5}, \frac{1}{6}$. ### Common Pitfall Be careful when comparing fractions. Students sometimes assume that a larger denominator implies a larger number, mistakenly choosing $\frac{1}{6}$ over $\frac{2}{5}$. Always convert to decimals or cross-multiply to verify. ### Final Answer Therefore, the correct answer is **$\frac{9}{4}, \frac{2}{5}$**.