If x= 2 is a factor of `x^2+ ax + b` and a +b =1. find the values of a and b.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that \( x = 2 \) is a factor of the quadratic equation \( x^2 + ax + b \) and that \( a + b = 1 \).

### Step-by-Step Solution:

1. **Understanding the Factor Condition**:
Since \( x = 2 \) is a factor of \( x^2 + ax + b \), substituting \( x = 2 \) into the equation should yield 0:
\[ 2^2 + a(2) + b = 0 \]
This simplifies to:
\[ 4 + 2a + b = 0 \quad \text{(Equation 1)} \]

2. **Using the Given Condition**:
We also have the condition \( a + b = 1 \):
\[ a + b = 1 \quad \text{(Equation 2)} \]

3. **Substituting Equation 2 into Equation 1**:
From Equation 2, we can express \( b \) in terms of \( a \):
\[ b = 1 - a \]
Now substitute \( b \) into Equation 1:
\[ 4 + 2a + (1 - a) = 0 \]
Simplifying this gives:
\[ 4 + 2a + 1 - a = 0 \]
\[ 5 + a = 0 \]
Thus, we find:
\[ a = -5 \]

4. **Finding \( b \)**:
Now that we have \( a \), we can substitute it back into Equation 2 to find \( b \):
\[ -5 + b = 1 \]
Solving for \( b \):
\[ b = 1 + 5 = 6 \]

5. **Final Values**:
Therefore, the values of \( a \) and \( b \) are:
\[ a = -5, \quad b = 6 \]

### Summary:
The values are:
- \( a = -5 \)
- \( b = 6 \)