solve `0=1+2x+3x^2`

Options

A. `(-1/4,0)`
B. `(-11,3/4)`
C. `(-3/4,-1/2)`
D. `(0,1/4)`

Explanation

To solve the equation \(0 = 1 + 2x + 3x^2\), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to set it to zero:
\[ 3x^2 + 2x + 1 = 0 \]

### Step 2: Identifying Coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), we identify the coefficients:
- \(a = 3\)
- \(b = 2\)
- \(c = 1\)

### Step 3: Calculating the Discriminant
Next, we calculate the discriminant \(D\) using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = 2^2 - 4 \cdot 3 \cdot 1 = 4 - 12 = -8 \]

### Step 4: Analyzing the Discriminant
Since the discriminant \(D\) is negative (\(-8\)), this indicates that the quadratic equation has no real roots and instead has complex (imaginary) roots.

### Step 5: Applying the Quadratic Formula
We use the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting \(b = 2\), \(D = -8\), and \(a = 3\):
\[ x = \frac{-2 \pm \sqrt{-8}}{2 \cdot 3} \]

### Step 6: Simplifying the Expression
We know that \(\sqrt{-8} = \sqrt{8} \cdot i = 2\sqrt{2} \cdot i\). Thus, we can substitute this back into our equation:
\[ x = \frac{-2 \pm 2\sqrt{2}i}{6} \]

### Step 7: Further Simplifying
Now, we can simplify the expression:
\[ x = \frac{-2}{6} \pm \frac{2\sqrt{2}i}{6} = -\frac{1}{3} \pm \frac{\sqrt{2}}{3}i \]

### Final Result
The solutions to the equation \(0 = 1 + 2x + 3x^2\) are:
\[ x = -\frac{1}{3} + \frac{\sqrt{2}}{3}i \quad \text{and} \quad x = -\frac{1}{3} - \frac{\sqrt{2}}{3}i \]

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