Suppose `a, b, c in R, a ne 0. If a + |b| + 2c = 0`, then roots of `ax^(2) + bx + c = 0` are

Options

A. real and distinct
B. real and equal
C. purely imaginary
D. non-real complex numbers

Answer

To solve the problem step by step, we start with the given equation and analyze it systematically.

### Step 1: Analyze the given equation
We are given the equation:
\[ a + |b| + 2c = 0 \]

### Step 2: Express |b| in terms of a and c
From the equation, we can isolate |b|:
\[ |b| = - (a + 2c) \]

### Step 3: Square both sides
Since |b| is always non-negative, we can square both sides:
\[ b^2 = (- (a + 2c))^2 \]
This simplifies to:
\[ b^2 = (a + 2c)^2 \]

### Step 4: Expand the squared term
Now, we expand the right-hand side:
\[ b^2 = a^2 + 4c^2 + 4ac \]

### Step 5: Write the discriminant of the quadratic equation
The quadratic equation we are considering is:
\[ ax^2 + bx + c = 0 \]
The discriminant \( D \) of this quadratic equation is given by:
\[ D = b^2 - 4ac \]

### Step 6: Substitute for b^2 in the discriminant
We substitute our expression for \( b^2 \) into the discriminant:
\[ D = (a^2 + 4c^2 + 4ac) - 4ac \]
This simplifies to:
\[ D = a^2 + 4c^2 \]

### Step 7: Analyze the discriminant
Since \( a^2 \) and \( 4c^2 \) are both non-negative (as squares of real numbers), we conclude that:
\[ D \geq 0 \]
However, \( D \) cannot be zero unless both \( a^2 = 0 \) and \( 4c^2 = 0 \). Since \( a \neq 0 \), \( D \) must be strictly positive.

### Step 8: Conclusion about the roots
Since the discriminant \( D \) is positive, the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real and distinct.

### Final Answer
The roots of the equation \( ax^2 + bx + c = 0 \) are real and distinct.

---