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 $5$. $\frac{7P}{P - 24} \times 0.2P - 91$ is equal to

### Concept & Algebraic Substitution To find the value of the given expression, we must first determine the value of the unknown constant $P$. We can find $P$ by utilizing the fact that $x=5$ is a root of equation A. $$ ax^2 + bx + c = 0 $$ ### Step-by-Step Solution 1. **Simplify Equation A:** $(x-2)^2 = -3x^2 + 2^2 + 25x - P$ $x^2 - 4x + 4 = -3x^2 + 4 + 25x - P$ Bring all terms to one side: $4x^2 - 29x + P = 0$ 2. **Find the value of $P$:** Since $x = 5$ is a root, substitute $x = 5$ into the simplified equation: $4(5)^2 - 29(5) + P = 0$ $100 - 145 + P = 0$ $-45 + P = 0 \implies P = 45$ 3. **Evaluate the target expression:** Substitute $P = 45$ into $\frac{7P}{P - 24} \times 0.2P - 91$: $= \frac{7(45)}{45 - 24} \times 0.2(45) - 91$ $= \frac{315}{21} \times 9 - 91$ $= 15 \times 9 - 91$ $= 135 - 91 = 44$ 4. **Check the options by substituting $P = 45$:** (a) $P + 2 = 47$ (b) $2P + 2 = 92$ (c) $2P - 1 = 89$ (d) $P - 1 = 44$ The evaluated expression matches option (d). ### Exam Strategy & Shortcut Whenever a root is given for an equation with a single unknown variable, substituting the root directly is the fastest way to isolate the variable. Simplify expressions early to prevent arithmetic errors with large numbers. ### Common Pitfall A common mistake is failing to simplify equation A correctly before substitution, particularly forgetting to move the $-3x^2$ over to make it $4x^2$, which will result in an incorrect $P$ value. ### Final Answer Therefore, the correct answer is **P - 1**.