If $3x + 7 = x^2 + P = 7x + 5$, what is the value of $P$?

### Concept & Strategy This problem involves a continuous equation (an equality with three parts). The strategy is to split it into two manageable equations. First, solve for $x$ using the two linear parts, and then substitute that value into the quadratic part to solve for $P$. ### Step-by-Step Solution Given the continuous equality: $3x + 7 = x^2 + P = 7x + 5$ **Step 1: Solve for $x$** Take the first and third parts of the equation, as they only contain one variable ($x$): $3x + 7 = 7x + 5$ Rearrange to isolate $x$: $7 - 5 = 7x - 3x$ $2 = 4x$ $x = \frac{2}{4} = \frac{1}{2}$ **Step 2: Solve for $P$** Now, use the second and third parts (or first and second) and substitute $x = 0.5$: $x^2 + P = 7x + 5$ Substitute $x = \frac{1}{2}$: $(\frac{1}{2})^2 + P = 7(\frac{1}{2}) + 5$ $\frac{1}{4} + P = 3.5 + 5$ $0.25 + P = 8.5$ Isolate $P$: $P = 8.5 - 0.25$ $P = 8.25$ Convert the decimal back to a mixed fraction to match the options: $8.25 = 8 \frac{1}{4}$ ### Exam Strategy & Shortcut **Direct Substitution:** You can quickly solve $3x + 7 = 7x + 5$ mentally. The difference in $x$ is 4, the difference in constants is 2. So $x = 2/4 = 0.5$. Plug $0.5$ into the easiest side: $3(0.5) + 7 = 8.5$. Now you know $x^2 + P = 8.5$. Since $x^2 = 0.25$, $P$ must be $8.5 - 0.25 = 8.25$. This avoids writing down multiple lines of algebra. ### Common Pitfall Students often get intimidated by the double equals sign and try to set up complex quadratic formulas. Always look for the easiest path first—in this case, equating the linear ends completely bypasses the need for quadratic factoring. ### Final Answer Therefore, the correct answer is **$8 \frac{1}{4}$**.