Questions: 6r^2 + 6r + 12 = 0

Transcript text: $6 r^{2}+6 r+12=0$

Solution

Step 1: Define the Polynomial Equation

We start with the polynomial equation given by:

\[ 6r^{2} + 6r + 12 = 0 \]

Step 2: Factor the Polynomial

The polynomial can be factorized as follows:

\[ 6 \left(r^{2} + r + 2\right) \]

Step 3: Solve the Polynomial Equation

To find the solutions for the equation, we solve:

\[ 6 \left(r^{2} + r + 2\right) = 0 \]

This leads us to the quadratic equation $ r^{2} + r + 2 = 0 $. The solutions to this equation are:

\[ r = -\frac{1}{2} - \frac{\sqrt{7}}{2}i \quad \text{and} \quad r = -\frac{1}{2} + \frac{\sqrt{7}}{2}i \]

The solutions to the polynomial equation are:

\[ \boxed{r = -\frac{1}{2} - \frac{\sqrt{7}}{2}i \quad \text{and} \quad r = -\frac{1}{2} + \frac{\sqrt{7}}{2}i} \]

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