Let a and b be two integers such that `10 a + b = 20 and g(x) = x^2 + ax + b.` If `g(10) g(11) = g(n),` then n is equal to

Options

A. 115
B. 120
C. 130
D. 140

Answer

To solve the problem, we need to follow these steps:

### Step 1: Set up the equations
We are given two equations:
1. \( 10a + b = 20 \)
2. \( g(x) = x^2 + ax + b \)

### Step 2: Find values for \( a \) and \( b \)
From the first equation, we can express \( b \) in terms of \( a \):
\[ b = 20 - 10a \]

### Step 3: Substitute \( a \) and \( b \) into \( g(x) \)
Now, substituting \( b \) into the function \( g(x) \):
\[ g(x) = x^2 + ax + (20 - 10a) \]
This simplifies to:
\[ g(x) = x^2 + ax + 20 - 10a \]

### Step 4: Calculate \( g(10) \) and \( g(11) \)
Now we calculate \( g(10) \) and \( g(11) \):
1. For \( g(10) \):
\[ g(10) = 10^2 + 10a + (20 - 10a) = 100 + 10a + 20 - 10a = 120 \]
2. For \( g(11) \):
\[ g(11) = 11^2 + 11a + (20 - 10a) = 121 + 11a + 20 - 10a = 141 + a \]

### Step 5: Set up the equation \( g(10) \cdot g(11) = g(n) \)
We know that:
\[ g(10) \cdot g(11) = g(n) \]
Substituting the values we found:
\[ 120 \cdot (141 + a) = g(n) \]

### Step 6: Find \( g(n) \)
Now, we express \( g(n) \):
\[ g(n) = n^2 + an + (20 - 10a) \]
This gives us:
\[ g(n) = n^2 + an + 20 - 10a \]

### Step 7: Set the equation
Now we equate the two expressions:
\[ 120 \cdot (141 + a) = n^2 + an + 20 - 10a \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ n^2 + an + 20 - 10a - 120 \cdot (141 + a) = 0 \]

### Step 9: Simplify the equation
Calculating \( 120 \cdot (141 + a) \):
\[ 120 \cdot 141 = 16920 \]
So, the equation simplifies to:
\[ n^2 + an + 20 - 10a - 16920 - 120a = 0 \]
\[ n^2 + an - 130a - 16900 = 0 \]

### Step 10: Solve for \( n \)
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here \( a = 1, b = a, c = -130a - 16900 \):
1. Calculate the discriminant:
\[ D = a^2 - 4 \cdot 1 \cdot (-130a - 16900) = a^2 + 520a + 67600 \]
2. Substitute into the quadratic formula:
\[ n = \frac{-a \pm \sqrt{D}}{2} \]

### Step 11: Find the value of \( n \)
Assuming \( a = 2 \) (since \( 10a + b = 20 \) gives \( b = 0 \)):
1. Substitute \( a = 2 \):
\[ D = 2^2 + 520 \cdot 2 + 67600 = 4 + 1040 + 67600 = 68644 \]
2. Calculate \( n \):
\[ n = \frac{-2 \pm \sqrt{68644}}{2} \]
Since \( \sqrt{68644} = 262 \):
\[ n = \frac{-2 \pm 262}{2} \]
This gives two possible values:
\[ n = \frac{260}{2} = 130 \quad \text{(positive solution)} \]
\[ n = \frac{-264}{2} = -132 \quad \text{(negative solution)} \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{130} \]