Let's analyze the given quadratic equations to determine whether they have two real solutions, one real solution, or two imaginary solutions. This can be done by calculating the discriminant $\Delta$ for each equation.

For a quadratic equation of the form:

$ax^2 + bx + c = 0$

The discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac$

• If $\Delta > 0$, there are two distinct real solutions.
• If $\Delta = 0$, there is one real solution.
• If $\Delta < 0$, there are two imaginary (complex) solutions.

Let’s solve each equation step by step.

11. $3x^2 - 240 = 0$

Rewrite as: $3x^2 = 240$ This is a standard quadratic equation, and we can directly solve for the discriminant:

• $a = 3$
• $b = 0$
• $c = -240$

Discriminant $\Delta = 0^2 - 4(3)(-240) = 2880$

Since $\Delta > 0$, two real solutions.

12. $-5x^2 = 60$

Rewrite as: $-5x^2 - 60 = 0$

• $a = -5$
• $b = 0$
• $c = -60$

Discriminant $\Delta = 0^2 - 4(-5)(-60) = -1200$

Since $\Delta < 0$, two imaginary solutions.

13. $\frac{x^2}{4} + 9 = 0$

Rewrite as: $\frac{x^2}{4} = -9$ or $x^2 = -36$

• This gives imaginary solutions directly because $x^2 = -36$.

Thus, two imaginary solutions.

14. $7(x + 1)^2 - 24 = 0$

Rewrite as: $7(x + 1)^2 = 24$, or $(x + 1)^2 = \frac{24}{7}$

• Taking the square root, we find two real roots.

Thus, two real solutions.

15. $2(3x^2 + 16) = 32$

Simplify to: $3x^2 + 16 = 16$, then $3x^2 = 0$

• This simplifies to $x = 0$.

Thus, one real solution.

16. $-\frac{x^2}{2} + 12 = 0$

Rewrite as: $-\frac{x^2}{2} = -12$, or $x^2 = 24$

• This gives two real roots since $x = \pm \sqrt{24}$.

Thus, two real solutions.

Summary:

• 11: Two real solutions
• 12: Two imaginary solutions
• 13: Two imaginary solutions
• 14: Two real solutions
• 15: One real solution
• 16: Two real solutions

Would you like more details or have any questions about these calculations?

Here are five related questions:

1. What is the significance of the discriminant in quadratic equations?
2. How can you factor a quadratic equation to find its roots?
3. Can a quadratic equation have both real and imaginary solutions?
4. How do you solve a quadratic equation by completing the square?
5. What is the quadratic formula and when is it used?

Tip: When solving quadratic equations, always check if they can be factored easily before using the quadratic formula!