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 step by step, let's follow the reasoning presented in the video transcript:

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

### Step 2: Identify the roots of the new equation
The roots of the equation \( x^2 + 4x + 1 = 0 \) can be denoted as \( \alpha \) and \( \beta \). According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -4 \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = 1 \)

Since the coefficient of \( x^2 \) is 1, we have:
- \( \alpha + \beta = -4 \)
- \( \alpha \beta = 1 \)

### Step 3: Relate the roots of the original equation
If the roots of the original equation \( ax^2 + bx + c = 0 \) are \( \alpha + 1 \) and \( \beta + 1 \), then:
- The sum of the roots of the original equation is:
\[ (\alpha + 1) + (\beta + 1) = \alpha + \beta + 2 = -4 + 2 = -2 \]
- The product of the roots of the original equation is:
\[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 = 1 - 4 + 1 = -2 \]

### Step 4: Use Vieta's formulas for the original equation
From Vieta's formulas for the original equation \( ax^2 + bx + c = 0 \):
- The sum of the roots is given by:
\[ -\frac{b}{a} = -2 \implies b = 2a \]
- The product of the roots is given by:
\[ \frac{c}{a} = -2 \implies c = -2a \]

### Step 5: Calculate \( b + c \)
Now we can find \( b + c \):
\[ b + c = 2a - 2a = 0 \]

### Conclusion
The numerical value of \( b + c \) is \( 0 \).