The point of intersection of the lines `2x^(2)-5xy+3y^(2)+8x-9y+6=0` is

Options

A. `(-3,4)`
B. `(3,-5)`
C. `(3,4)`
D. `(-3,-5)`

Explanation

To find the point of intersection of the lines given by the equation \(2x^2 - 5xy + 3y^2 + 8x - 9y + 6 = 0\), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ 2x^2 - 5xy + 3y^2 + 8x - 9y + 6 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic in \(x\) and \(y\). To do this, we can rearrange the equation:
\[ 2x^2 - 5xy + 3y^2 + 8x - 9y + 6 = 0 \]
We can group the terms involving \(x\):
\[ 2x^2 - (5y - 8)x + (3y^2 - 9y + 6) = 0 \]

### Step 3: Use the quadratic formula
We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find \(x\) in terms of \(y\):
- Here, \(a = 2\), \(b = -(5y - 8)\), and \(c = 3y^2 - 9y + 6\).

### Step 4: Calculate the discriminant
First, we need to calculate the discriminant \(D = b^2 - 4ac\):
\[ D = (5y - 8)^2 - 4 \cdot 2 \cdot (3y^2 - 9y + 6) \]
Simplifying this will help us find the values of \(y\).

### Step 5: Solve for \(y\)
After calculating \(D\), we can set the discriminant equal to zero to find the values of \(y\) that give us the intersection points:
\[ D = 0 \]

### Step 6: Substitute \(y\) back to find \(x\)
Once we have the values of \(y\), we can substitute them back into the equation to find the corresponding \(x\) values.

### Step 7: Write the point of intersection
The point of intersection will be given by the coordinates \((x, y)\).

### Final Result
After performing the calculations, we find that the point of intersection is:
\[ (3, 4) \]