If a,b,c are real distinct numbers such that `a ^(3) +b ^(3) +c ^(3)= 3abc,` then the quadratic equation `ax ^(2) +bx +c =0` has (a) Real roots (b) At least one negative root (c) Both roots are negative (d) Non real roots

Options

A. Real roots
B. At least one negative root
C. Both roots are negative
D. Non real roots

Answer

To solve the problem, we start with the given condition and analyze the quadratic equation.

### Step 1: Understand the given condition
We are given that \( a^3 + b^3 + c^3 = 3abc \). This is a specific case that can be rewritten using a well-known identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]
From this identity, we can conclude that:
\[ a + b + c = 0 \]
since \( a, b, c \) are distinct real numbers.

**Hint:** Recognize that the identity relates the sum of cubes to the sum of the numbers and their products.

### Step 2: Analyze the quadratic equation
The quadratic equation we need to analyze is:
\[ ax^2 + bx + c = 0 \]
Given that \( a + b + c = 0 \), we can substitute \( c = - (a + b) \) into the equation:
\[ ax^2 + bx - (a + b) = 0 \]

**Hint:** Substitute \( c \) to simplify the quadratic equation.

### Step 3: Find the roots of the quadratic equation
Now, we can apply the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4a(- (a + b))}}{2a} \]
This simplifies to:
\[ x = \frac{-b \pm \sqrt{b^2 + 4a(a + b)}}{2a} \]

**Hint:** Use the quadratic formula to find the roots.

### Step 4: Determine the nature of the roots
To analyze the roots further, we need to check the discriminant:
\[ D = b^2 + 4a(a + b) \]
Since \( a + b + c = 0 \), we can conclude that \( D \) is positive because \( b^2 \) is always non-negative, and \( 4a(a + b) \) will also be non-negative if \( a \) and \( b \) are distinct real numbers.

**Hint:** Check the discriminant to determine if the roots are real.

### Step 5: Check the sign of the roots
Since \( a + b + c = 0 \) implies that one root is positive and the other root is negative (because the sum of the roots is zero), we can conclude that:
- The quadratic equation has real roots.
- At least one root is negative.

**Hint:** Analyze the sum of the roots to determine their signs.

### Conclusion
Based on the analysis, the correct option is:
(a) Real roots