If 8, 2 are roots of the equation` x^2 + ax + beta `and 3, 3 are roots of `x^2 + alpha x + b = 0` then roots of the equation `x^2+ax+b =0` are

Options

A. `8, -1`
B. `-9, 2`
C. `-8, -2`
D. 9, 1

Answer

To solve the problem step-by-step, we will use the information given about the roots of the equations and apply the relationships between coefficients and roots.

### Step 1: Identify the first equation and its roots
The first equation is given as:
\[ x^2 + ax + \beta = 0 \]
with roots 8 and 2.

Using the relationships for the sum and product of roots:
- Sum of roots \( = 8 + 2 = 10 \)
- Product of roots \( = 8 \times 2 = 16 \)

From Vieta's formulas:
- The sum of the roots is equal to \(-a\), so:
\[ -a = 10 \implies a = -10 \]
- The product of the roots is equal to \(\beta\), so:
\[ \beta = 16 \]

### Step 2: Identify the second equation and its roots
The second equation is given as:
\[ x^2 + \alpha x + b = 0 \]
with roots 3 and 3 (a repeated root).

Using the relationships for the sum and product of roots:
- Sum of roots \( = 3 + 3 = 6 \)
- Product of roots \( = 3 \times 3 = 9 \)

From Vieta's formulas:
- The sum of the roots is equal to \(-\alpha\), so:
\[ -\alpha = 6 \implies \alpha = -6 \]
- The product of the roots is equal to \(b\), so:
\[ b = 9 \]

### Step 3: Form the third equation
Now we have:
- \(a = -10\)
- \(b = 9\)

We can substitute these values into the equation:
\[ x^2 + ax + b = 0 \]
This becomes:
\[ x^2 - 10x + 9 = 0 \]

### Step 4: Solve the third equation
To find the roots of the equation \(x^2 - 10x + 9 = 0\), we can factor it:
\[ x^2 - 10x + 9 = (x - 1)(x - 9) = 0 \]

Setting each factor to zero gives us the roots:
\[ x - 1 = 0 \implies x = 1 \]
\[ x - 9 = 0 \implies x = 9 \]

### Conclusion
The roots of the equation \(x^2 + ax + b = 0\) are:
\[ \boxed{1 \text{ and } 9} \]