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

Options

A. (a) 76
B. (b) 100
C. (c) 294
D. (d) 318

Answer

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

### Step 1: Solve the first equation
The first equation is:
\[ x^2 - 8x + 15 = 0 \]

To factor this equation, we look for two numbers that multiply to 15 and add up to -8. These numbers are -5 and -3. Thus, we can factor the equation as:
\[ (x - 5)(x - 3) = 0 \]

Setting each factor to zero gives us the roots:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

So, the roots of the first equation are \( x = 5 \) and \( x = 3 \).

### Step 2: Solve the second equation
The second equation is:
\[ y^2 + 2y - 35 = 0 \]

To factor this equation, we look for two numbers that multiply to -35 and add up to 2. These numbers are 7 and -5. Thus, we can factor the equation as:
\[ (y + 7)(y - 5) = 0 \]

Setting each factor to zero gives us the roots:
\[ y + 7 = 0 \quad \Rightarrow \quad y = -7 \]
\[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \]

So, the roots of the second equation are \( y = -7 \) and \( y = 5 \).

### Step 3: Identify the common root
The common root between the two equations is:
\[ 5 \]

### Step 4: Calculate the cube and square of the common root
Now we need to calculate the cube and square of the common root (5):
- The square of 5 is:
\[ 5^2 = 25 \]
- The cube of 5 is:
\[ 5^3 = 125 \]

### Step 5: Find the difference between the cube and square
Now we find the difference between the cube and the square:
\[ 5^3 - 5^2 = 125 - 25 = 100 \]

### Final Answer
The difference of the cube and square of the common root is:
\[ \boxed{100} \]

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