https://file.aiquickdraw.com/mathbot/file/1727023747543aq93oz71.jpg

The image contains the equation:

\[
x^2 - 4x + 3y^2 = -2
\]

The task is to "Gambarkan grafik persamaan" which means "Draw the graph of the equation" in Indonesian.

Let's analyze and simplify this equation step by step.

### Step 1: Rearrange the equation
We start with:
\[
x^2 - 4x + 3y^2 = -2
\]
We want to transform this into a more recognizable form, such as for a conic section (ellipse, hyperbola, etc.).

### Step 2: Complete the square for the \(x\)-terms
The \(x\)-terms are \(x^2 - 4x\). To complete the square:
\[
x^2 - 4x = (x - 2)^2 - 4
\]

Substitute this into the original equation:
\[
(x - 2)^2 - 4 + 3y^2 = -2
\]
Now, move the constant \(-4\) to the right-hand side:
\[
(x - 2)^2 + 3y^2 = 2
\]

### Step 3: Recognize the form
This is now an equation of an ellipse:
\[
\frac{(x - 2)^2}{2} + \frac{y^2}{\frac{2}{3}} = 1
\]

This represents an ellipse centered at \((2, 0)\), with semi-major axis along the \(x\)-axis and semi-minor axis along the \(y\)-axis.

### Conclusion
The equation describes an ellipse centered at \((2, 0)\), with axes lengths scaled by \(\sqrt{2}\) along the \(x\)-axis and \(\frac{\sqrt{2}}{\sqrt{3}}\) along the \(y\)-axis. You can now draw the graph based on these parameters.

Would you like further details, or any clarification?

### Related Questions:
1. What is the general form of an ellipse equation?
2. How do you complete the square for more complex equations?
3. Can you explain the steps to identify different conic sections?
4. How does scaling affect the shape of an ellipse?
5. What transformations move the center of a conic section?

### Tip:
When identifying conic sections, completing the square is often a crucial step to simplify and recognize the equation.

Gambarkan grafik persamaan x^2 - 4x + 3y^2 = -2.

Explore how to graph the ellipse defined by the equation x^2 - 4x + 3y^2 = -2. This detailed step-by-step solution includes completing the square, recognizing conic sections, and identifying ellipses. Suitable for high school students in Grades 10-12.

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