how many real number solutions does the equation 3x^2+2x+6=0 have?

To determine the number of real solutions of the quadratic equation:

\[
3x^2 + 2x + 6 = 0
\]

we use the discriminant, \( \Delta \), which is part of the quadratic formula:

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

where the discriminant \( \Delta \) is given by:

\[
\Delta = b^2 - 4ac
\]

For the equation \( 3x^2 + 2x + 6 = 0 \):

- \(a = 3\)
- \(b = 2\)
- \(c = 6\)

Now, we calculate the discriminant:

\[
\Delta = (2)^2 - 4 \cdot 3 \cdot 6
\]

\[
\Delta = 4 - 72
\]

\[
\Delta = -68
\]

Since the discriminant \( \Delta \) is negative (\( -68 \)), there are no real solutions to the equation. A negative discriminant indicates that the solutions are complex (non-real).

### Conclusion:

The equation \(3x^2 + 2x + 6 = 0\) has **no real number solutions**.

---

Would you like more details or have any other questions? Here are some related questions:

1. How do you find complex solutions for a quadratic equation with a negative discriminant?
2. What is the geometric interpretation of a quadratic equation with no real solutions?
3. Can you explain how the discriminant determines the number and type of solutions?
4. How do quadratic equations apply to real-world problems?
5. What are the steps to graph a quadratic equation?

**Tip:** When dealing with quadratic equations, always check the discriminant first to determine the nature of the solutions.

Explore how to calculate the number of real solutions for the quadratic equation 3x^2 + 2x + 6 = 0 using the discriminant method. Understand the concept of discriminant in quadratic equations and its significance in determining the nature of solutions. Suitable for students in Grades 10-12.

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