If a and P are the roots of the equation `x^2-11x + 24= 0`, find the equation having the roots `a + 2 and P +2`

Options

A. `x^(2) + 15x+ 24= 0`
B. `x^(2)-15x+ 24=0`
C. `x^(2)+15x-50=0`
D. `x^(2)-15x+ 50=0`

Answer

To find the equation whose roots are \( a + 2 \) and \( p + 2 \), we start with the given quadratic equation:

\[ x^2 - 11x + 24 = 0 \]

### Step 1: Identify the roots
The roots of the equation are \( a \) and \( p \). By Vieta's formulas, we know:

- The sum of the roots \( a + p = 11 \)
- The product of the roots \( a \cdot p = 24 \)

### Step 2: Find the new roots
We need to find the new roots, which are \( a + 2 \) and \( p + 2 \).

### Step 3: Calculate the sum of the new roots
The sum of the new roots is:

\[ (a + 2) + (p + 2) = (a + p) + 4 = 11 + 4 = 15 \]

### Step 4: Calculate the product of the new roots
The product of the new roots is:

\[ (a + 2)(p + 2) = ap + 2a + 2p + 4 \]

Substituting the known values:

\[ = 24 + 2(a + p) + 4 = 24 + 2 \cdot 11 + 4 = 24 + 22 + 4 = 50 \]

### Step 5: Form the new quadratic equation
Using the sum and product of the new roots, we can form the new quadratic equation. The standard form of a quadratic equation with roots \( r_1 \) and \( r_2 \) is:

\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]

Substituting the values we found:

\[ x^2 - 15x + 50 = 0 \]

### Final Answer
The equation having the roots \( a + 2 \) and \( p + 2 \) is:

\[ x^2 - 15x + 50 = 0 \]

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