Each root of the equation `ax^2 + bx + c = 0` is decreased by 1. The quadratic equation with these roots is `x^2 + 4x + 1 = 0`. The numerical value of b + c is……………….

Answer

To solve the problem, we will start by analyzing the given information and using the properties of roots of quadratic equations.

### Step 1: Understand the given equations
We are given that each root of the equation \( ax^2 + bx + c = 0 \) is decreased by 1, resulting in the roots of the new equation \( x^2 + 4x + 1 = 0 \).

### Step 2: Identify the roots of the new equation
The roots of the equation \( x^2 + 4x + 1 = 0 \) can be found using the sum and product of roots:
- The sum of the roots \( \alpha + \beta = -\text{coefficient of } x = -4 \)
- The product of the roots \( \alpha \beta = \text{constant term} = 1 \)

### Step 3: Relate the roots of the original equation
If the roots of the original equation \( ax^2 + bx + c = 0 \) are \( \alpha \) and \( \beta \), then the roots of the new equation are \( \alpha - 1 \) and \( \beta - 1 \). Therefore, we can express the sum and product of the new roots:
- Sum of the new roots:
\[ (\alpha - 1) + (\beta - 1) = \alpha + \beta - 2 = -4 - 2 = -6 \]
- Product of the new roots:
\[ (\alpha - 1)(\beta - 1) = \alpha \beta - (\alpha + \beta) + 1 = 1 - (-4) + 1 = 1 + 4 + 1 = 6 \]

### Step 4: Set up the equations for the original roots
From the properties of the original equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 5: Set up the equations based on the new roots
From the new roots, we have:
1. \(-\frac{b}{a} - 2 = -6\) (from the sum of the new roots)
2. \(\frac{c}{a} = 6\) (from the product of the new roots)

### Step 6: Solve for \( b \) and \( c \)
From the first equation:
\[ -\frac{b}{a} - 2 = -6 \implies -\frac{b}{a} = -4 \implies \frac{b}{a} = 4 \implies b = 4a \]

From the second equation:
\[ \frac{c}{a} = 6 \implies c = 6a \]

### Step 7: Find \( b + c \)
Now we can find \( b + c \):
\[ b + c = 4a + 6a = 10a \]

### Step 8: Determine the numerical value of \( b + c \)
Since \( a \) is a non-zero constant, the numerical value of \( b + c \) can be expressed as:
\[ b + c = 10a \]
However, we need the numerical value, and since \( a \) can be any non-zero value, we can set \( a = 1 \) for simplicity:
\[ b + c = 10 \cdot 1 = 10 \]

Thus, the numerical value of \( b + c \) is **10**.