If both the roots of the equation `ax^(2) + bx + c = 0` are zero, then

Options

A. b = c = 0
B. `b = 0, c ne 0`
C. `b ne 0, c = 0`
D. none of these

Answer

To solve the problem, we need to analyze the quadratic equation given by \( ax^2 + bx + c = 0 \) under the condition that both roots are zero.

### Step 1: Understanding the Roots
If both roots of the equation are zero, it implies that the equation can be expressed in the form:
\[ a(x - 0)(x - 0) = 0 \]
This simplifies to:
\[ ax^2 = 0 \]
This means that the equation has a double root at \( x = 0 \).

### Step 2: Setting Up the Equation
From the standard form of the quadratic equation, we have:
\[ ax^2 + bx + c = 0 \]
Since both roots are zero, we can substitute \( x = 0 \) into the equation:
\[ a(0)^2 + b(0) + c = 0 \]
This simplifies to:
\[ c = 0 \]

### Step 3: Analyzing the Coefficient \( b \)
Next, we need to analyze the coefficient \( b \). The general formula for the roots of a quadratic equation is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Since both roots are zero, we can set the expression equal to zero:
\[ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = 0 \]
For this to hold true, the numerator must also be zero:
\[ -b = 0 \implies b = 0 \]

### Step 4: Conclusion
From the analysis, we have determined that:
- \( b = 0 \)
- \( c = 0 \)

Thus, both \( b \) and \( c \) must be equal to zero for the quadratic equation \( ax^2 + bx + c = 0 \) to have both roots as zero.

### Final Answer
Therefore, the values of \( b \) and \( c \) are both equal to zero.

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