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You've provided equations from an image that need solving. Let's solve each equation one at a time.

### Equation 1:
\[ y^2 - 2x^2 - 12x + 18y + 100 = 0 \]

### Equation 2:
\[ 3x^2 - y^2 - 12 = 0 \]

### Equation 3:
\[ 2x^2 - 3y^2 - 8x + 12y + 14 = 0 \]

Let's start by solving each of these equations. We'll handle them one by one to find any specific points or solutions. We'll begin with Equation 1 and proceed to the others.Here are the solutions for each equation in terms of \( y \) as a function of \( x \):

### Equation 1:
\[ y = -\sqrt{2x^2 + 12x - 19} - 9 \quad \text{or} \quad y = \sqrt{2x^2 + 12x - 19} - 9 \]

### Equation 2:
\[ y = -\sqrt{3x^2 - 12} \quad \text{or} \quad y = \sqrt{3x^2 - 12} \]

### Equation 3:
\[ y = 2 - \frac{\sqrt{6x^2 - 24x + 78}}{3} \quad \text{or} \quad y = \frac{\sqrt{6x^2 - 24x + 78}}{3} + 2 \]

These are the expressions for \( y \) as functions of \( x \) for each equation. If you need to find specific points or intersections, please specify further or ask for a graphical analysis. Would you like to proceed with any specific calculations or have further questions on these equations?

Here are some related questions you might find interesting:
1. How can you find the intersections of these equations?
2. What are the graphical representations of these equations?
3. How can you determine the range of \( x \) values for which these solutions are real?
4. Could these equations represent conic sections, and if so, what types?
5. What is the significance of the discriminant in each solution?

**Tip:** When dealing with square roots in equations, ensure the expressions under the square roots are non-negative to maintain real solutions.

Solve the system of equations: 
1. y^2 - 2x^2 - 12x + 18y + 100 = 0 
2. 3x^2 - y^2 - 12 = 0 
3. 2x^2 - 3y^2 - 8x + 12y + 14 = 0

Explore how to solve the system of quadratic equations involving variables x and y. Learn step-by-step techniques including solving, factoring, and the use of quadratic formula. Suitable for students in Grades 10-12.

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