Let `f(x)` be a quadratic expression such that `f(-1)+f(2)=0`. If one root of `f(x)=0` is `3`, then the other root of `f(x)=0` lies in (A) `(-oo,-3)` (B) `(-3,oo)` (C) `(0,5)` (D) `(5,oo)`

Options

A. `(-oo,-3)`
B. `(-3,oo)`
C. `(0,5)`
D. `(5,oo)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the quadratic function
Let \( f(x) = a(x - 3)(x - r) \), where \( r \) is the other root we need to find.

### Step 2: Use the given condition
We know that \( f(-1) + f(2) = 0 \). We will calculate \( f(-1) \) and \( f(2) \).

1. Calculate \( f(-1) \):
\[ f(-1) = a(-1 - 3)(-1 - r) = a(-4)(-1 - r) = 4a(1 + r) \]

2. Calculate \( f(2) \):
\[ f(2) = a(2 - 3)(2 - r) = a(-1)(2 - r) = -a(2 - r) \]

### Step 3: Set up the equation
Now we substitute these into the given condition:
\[ f(-1) + f(2) = 0 \]
This gives us:
\[ 4a(1 + r) - a(2 - r) = 0 \]

### Step 4: Simplify the equation
Factor out \( a \) (assuming \( a \neq 0 \)):
\[ 4(1 + r) - (2 - r) = 0 \]
Expanding this:
\[ 4 + 4r - 2 + r = 0 \]
Combine like terms:
\[ 5r + 2 = 0 \]
Thus,
\[ 5r = -2 \implies r = -\frac{2}{5} \]

### Step 5: Determine the range of the other root
Now we have the roots of the quadratic function:
1. One root is \( 3 \)
2. The other root is \( -\frac{2}{5} \)

### Step 6: Analyze the range of the other root
The root \( -\frac{2}{5} \) is approximately \( -0.4 \). We need to determine which interval this root lies in:
- \( -\frac{2}{5} \) is greater than \( -3 \) and less than \( 0 \).

### Conclusion
Thus, the other root lies in the interval \( (-3, 0) \).

### Final Answer
The correct option is:
(B) \( (-3, \infty) \)