If the factors of `5x^(2)-18x+9` are (ax+b), (x+b) then the value of a & b of are ...........& ..........respectively.

Answer

To find the values of \( a \) and \( b \) in the factors of the polynomial \( 5x^2 - 18x + 9 \) given as \( (ax + b)(x + b) \), we will follow these steps:

### Step 1: Expand the factors
We start by expanding the expression \( (ax + b)(x + b) \).

\[ (ax + b)(x + b) = ax^2 + abx + bx + b^2 \]

Combining like terms, we get:

\[ ax^2 + (ab + b)x + b^2 \]

### Step 2: Set the expanded expression equal to the original polynomial
Now we set the expanded expression equal to the original polynomial:

\[ ax^2 + (ab + b)x + b^2 = 5x^2 - 18x + 9 \]

### Step 3: Compare coefficients
From the equation above, we can compare the coefficients of \( x^2 \), \( x \), and the constant term:

1. Coefficient of \( x^2 \):
\[ a = 5 \]

2. Coefficient of \( x \):
\[ ab + b = -18 \]

3. Constant term:
\[ b^2 = 9 \]

### Step 4: Solve for \( b \)
From the constant term equation \( b^2 = 9 \), we can find \( b \):

\[ b = 3 \quad \text{or} \quad b = -3 \]

### Step 5: Substitute \( b \) back to find \( a \)
Now we substitute \( b \) into the equation \( ab + b = -18 \) to find \( a \):

1. If \( b = 3 \):
\[ a(3) + 3 = -18 \implies 3a + 3 = -18 \implies 3a = -21 \implies a = -7 \quad \text{(not valid since we already found } a = 5\text{)} \]

2. If \( b = -3 \):
\[ a(-3) - 3 = -18 \implies -3a - 3 = -18 \implies -3a = -15 \implies a = 5 \quad \text{(valid)} \]

### Conclusion
Thus, the values of \( a \) and \( b \) are:

\[ a = 5 \quad \text{and} \quad b = -3 \]

### Final Answer
The value of \( a \) is \( 5 \) and the value of \( b \) is \( -3 \).

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