The perimeter of a tringle is `15x^(2) - 23x + 9` and two of its sides are `5x^(2) - 8x -1` and `6x^(2) -9x + 4`. Find the side.

Options

A. `4x^(2) - 22x - 6`
B. `4x^(2) + 22x + 6`
C. `4x^(2) - 22x + 6`
D. `4x^(2) + 22x - 6`

Answer

To find the third side of the triangle, we can use the information given about the perimeter and the two sides. Here’s how to solve the problem step by step:

### Step 1: Write down the formula for the perimeter of a triangle.
The perimeter \( P \) of a triangle is the sum of the lengths of all its sides. If we denote the sides as \( a \), \( b \), and \( c \), then:
\[ P = a + b + c \]

### Step 2: Identify the given values.
From the question, we know:
- The perimeter \( P = 15x^2 - 23x + 9 \)
- The first side \( a = 5x^2 - 8x - 1 \)
- The second side \( b = 6x^2 - 9x + 4 \)

### Step 3: Set up the equation to find the third side \( c \).
We can express the third side \( c \) as:
\[ c = P - (a + b) \]

### Step 4: Calculate \( a + b \).
First, we need to find \( a + b \):
\[ a + b = (5x^2 - 8x - 1) + (6x^2 - 9x + 4) \]
Combine like terms:
\[ = (5x^2 + 6x^2) + (-8x - 9x) + (-1 + 4) \]
\[ = 11x^2 - 17x + 3 \]

### Step 5: Substitute \( a + b \) into the equation for \( c \).
Now we can substitute \( a + b \) into the equation for \( c \):
\[ c = (15x^2 - 23x + 9) - (11x^2 - 17x + 3) \]

### Step 6: Distribute the negative sign and combine like terms.
Distributing the negative sign gives:
\[ c = 15x^2 - 23x + 9 - 11x^2 + 17x - 3 \]
Now combine like terms:
\[ = (15x^2 - 11x^2) + (-23x + 17x) + (9 - 3) \]
\[ = 4x^2 - 6x + 6 \]

### Final Answer:
Thus, the third side of the triangle is:
\[ c = 4x^2 - 6x + 6 \]