if `3x +7 = x^(2) + P = 7x + 5 ` What is the value of of P

Options

A. `(1)/(2)`
B. `8 (1)/(4)`
C. Can't be determined
D. None of these

Answer

To find the value of \( P \) in the equation \( 3x + 7 = x^2 + P = 7x + 5 \), we can follow these steps:

### Step 1: Set up the equations
We know that:
\[ 3x + 7 = 7x + 5 \]
This is because all three expressions are equal.

### Step 2: Rearrange the equation
Rearranging the equation gives:
\[ 3x + 7 - 7x - 5 = 0 \]
This simplifies to:
\[ -4x + 2 = 0 \]

### Step 3: Solve for \( x \)
Now, we can solve for \( x \):
\[ -4x = -2 \implies x = \frac{1}{2} \]

### Step 4: Substitute \( x \) back into one of the original equations
We can substitute \( x = \frac{1}{2} \) back into \( 3x + 7 \) to find the value of \( P \):
\[ 3\left(\frac{1}{2}\right) + 7 = \frac{3}{2} + 7 = \frac{3}{2} + \frac{14}{2} = \frac{17}{2} \]

### Step 5: Use the value of \( x \) to find \( P \)
Now we use the equation \( x^2 + P = \frac{17}{2} \):
\[ \left(\frac{1}{2}\right)^2 + P = \frac{17}{2} \]
This simplifies to:
\[ \frac{1}{4} + P = \frac{17}{2} \]

### Step 6: Solve for \( P \)
Now, we can isolate \( P \):
\[ P = \frac{17}{2} - \frac{1}{4} \]
To subtract these fractions, we need a common denominator, which is 4:
\[ P = \frac{34}{4} - \frac{1}{4} = \frac{34 - 1}{4} = \frac{33}{4} \]

Thus, the value of \( P \) is:
\[ \boxed{\frac{33}{4}} \]