Show that the point (2,3) lies outside the parabola `y^2=3x`.

Answer

To determine whether the point (2, 3) lies outside the parabola given by the equation \( y^2 = 3x \), we can follow these steps:

### Step 1: Rewrite the equation of the parabola
The equation of the parabola is given as:
\[ y^2 = 3x \]
We can rearrange this to:
\[ y^2 - 3x = 0 \]
Let’s denote this as \( t \):
\[ t = y^2 - 3x \]

### Step 2: Substitute the point into the equation
Now we will substitute the coordinates of the point (2, 3) into the equation:
\[ x = 2 \quad \text{and} \quad y = 3 \]
Substituting these values into \( t \):
\[ t = 3^2 - 3(2) \]

### Step 3: Calculate the value of \( t \)
Now, calculate \( t \):
\[ t = 9 - 6 = 3 \]

### Step 4: Analyze the value of \( t \)
We found that:
\[ t = 3 \]
Since \( t > 0 \), this indicates that the point (2, 3) lies outside the parabola.

### Conclusion
Therefore, we conclude that the point (2, 3) lies outside the parabola \( y^2 = 3x \).

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