What is the difference of the cube and square of the common root of `(x^(2)-8x + 15) = 0` and `( y ^(2) + 2y - 35 ) = 0` ?

Options

A. 76
B. 100
C. 294
D. 318

Answer

To solve the problem, we need to find the common root of the two quadratic equations given and then calculate the difference between the cube and square of that common root.

### Step-by-Step Solution:

1. **Identify the first equation**:
The first equation is \( x^2 - 8x + 15 = 0 \).

2. **Factor the first equation**:
We need to factor \( x^2 - 8x + 15 \). We look for two numbers that multiply to \( 15 \) and add up to \( -8 \).
- The numbers are \( -5 \) and \( -3 \).
- Thus, we can factor it as:
\[ (x - 5)(x - 3) = 0 \]

3. **Find the roots of the first equation**:
Setting each factor to zero gives us:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
So, the roots of the first equation are \( x = 5 \) and \( x = 3 \).

4. **Identify the second equation**:
The second equation is \( y^2 + 2y - 35 = 0 \).

5. **Factor the second equation**:
We need to factor \( y^2 + 2y - 35 \). We look for two numbers that multiply to \( -35 \) and add up to \( 2 \).
- The numbers are \( 7 \) and \( -5 \).
- Thus, we can factor it as:
\[ (y + 7)(y - 5) = 0 \]

6. **Find the roots of the second equation**:
Setting each factor to zero gives us:
\[ y + 7 = 0 \quad \Rightarrow \quad y = -7 \]
\[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \]
So, the roots of the second equation are \( y = -7 \) and \( y = 5 \).

7. **Identify the common root**:
The common root between the two equations is \( 5 \).

8. **Calculate the difference of the cube and square of the common root**:
We need to find \( 5^3 - 5^2 \).
- Calculate \( 5^3 \):
\[ 5^3 = 125 \]
- Calculate \( 5^2 \):
\[ 5^2 = 25 \]
- Now, find the difference:
\[ 5^3 - 5^2 = 125 - 25 = 100 \]

9. **Final Answer**:
The difference of the cube and square of the common root is \( 100 \).