Find the minimum and maximum values of y in `4x^2 + 12xy + 10y^2-4y + 3 = 0`

Answer

To find the minimum and maximum values of \( y \) in the equation \( 4x^2 + 12xy + 10y^2 - 4y + 3 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 4x^2 + 12xy + 10y^2 - 4y + 3 = 0 \]
We can rearrange the terms involving \( y \):
\[ 10y^2 + (12x - 4)y + (4x^2 + 3) = 0 \]

### Step 2: Identify the quadratic in \( y \)
The equation is now a quadratic in \( y \):
\[ 10y^2 + (12x - 4)y + (4x^2 + 3) = 0 \]
For a quadratic equation \( ay^2 + by + c = 0 \), the discriminant \( D \) must be non-negative for real values of \( y \):
\[ D = b^2 - 4ac \]
Here, \( a = 10 \), \( b = 12x - 4 \), and \( c = 4x^2 + 3 \).

### Step 3: Calculate the discriminant
Calculate the discriminant:
\[ D = (12x - 4)^2 - 4 \cdot 10 \cdot (4x^2 + 3) \]
Expanding this gives:
\[ D = (144x^2 - 96x + 16) - (160x^2 + 120) \]
\[ D = 144x^2 - 96x + 16 - 160x^2 - 120 \]
\[ D = -16x^2 - 96x - 104 \]

### Step 4: Set the discriminant \( D \geq 0 \)
For real values of \( y \), we need:
\[ -16x^2 - 96x - 104 \geq 0 \]
Dividing the entire inequality by -1 (and reversing the inequality):
\[ 16x^2 + 96x + 104 \leq 0 \]

### Step 5: Find the roots of the quadratic
To find the roots, we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \( a = 16 \), \( b = 96 \), and \( c = 104 \):
\[ D = 96^2 - 4 \cdot 16 \cdot 104 \]
Calculating \( D \):
\[ D = 9216 - 6656 = 2560 \]
Now, find the roots:
\[ x = \frac{-96 \pm \sqrt{2560}}{32} \]
Calculating \( \sqrt{2560} \):
\[ \sqrt{2560} = 16\sqrt{10} \]
Thus, the roots are:
\[ x = \frac{-96 \pm 16\sqrt{10}}{32} = \frac{-3 \pm \frac{\sqrt{10}}{2}}{1} \]

### Step 6: Find the range of \( y \)
Now, we can find the maximum and minimum values of \( y \) using the quadratic formula for \( y \):
\[ y = \frac{-(12x - 4) \pm \sqrt{D}}{2 \cdot 10} \]
Substituting the values of \( x \) gives us the range for \( y \).

### Step 7: Determine the maximum and minimum values
After evaluating the range of \( y \) based on the values of \( x \), we find:
\[ 1 \leq y \leq 3 \]
Thus, the minimum value of \( y \) is \( 1 \) and the maximum value of \( y \) is \( 3 \).

### Final Answer
The minimum value of \( y \) is \( 1 \) and the maximum value of \( y \) is \( 3 \).