Let's solve the given quadratic equation:

$3x^2 + 3x + 12 = 10$

Step 1: Rearrange the equation

First, move the constant term on the right side to the left side:

$3x^2 + 3x + 12 - 10 = 0$

Simplifying:

$3x^2 + 3x + 2 = 0$

Step 2: Apply the quadratic formula

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $3x^2 + 3x + 2 = 0$, the coefficients are:

• $a = 3$
• $b = 3$
• $c = 2$

Step 3: Calculate the discriminant

The discriminant is given by:

$\Delta = b^2 - 4ac = 3^2 - 4(3)(2) = 9 - 24 = -15$

Since the discriminant is negative ($\Delta = -15$), we will have complex (imaginary) solutions.

Step 4: Solve for $x$

Now, plug the values of $a$, $b$, and $\Delta$ into the quadratic formula:

$x = \frac{-3 \pm \sqrt{-15}}{6}$

We know that $\sqrt{-15} = i\sqrt{15}$, where $i$ is the imaginary unit. So, the solutions are:

$x = \frac{-3 \pm i\sqrt{15}}{6}$

Step 5: Simplify the solutions

The two solutions are:

$x = \frac{-3}{6} \pm \frac{i\sqrt{15}}{6}$

Simplifying:

$x = -\frac{1}{2} \pm \frac{i\sqrt{15}}{6}$

So the solutions are:

$x = -\frac{1}{2} \pm \frac{i\sqrt{15}}{6}$

This is the final answer in the form $a \pm bi$, where $a = -\frac{1}{2}$ and $b = \frac{\sqrt{15}}{6}$.

Let me know if you need further clarification or have more questions!

Additional related questions:

1. What happens if the discriminant is positive?
2. How do you simplify complex numbers in quadratic equations?
3. How can you graph quadratic functions with imaginary solutions?
4. How do you find the vertex of a quadratic function?
5. How can complex solutions be applied in real-world scenarios?

Tip:

When solving quadratic equations with complex solutions, it's helpful to remember that the discriminant ($b^2 - 4ac$) determines the type of solutions: positive means real solutions, zero means one real solution, and negative means complex solutions.