For real numbers x and y, if `x^2+xy-y^2+2x -y+1=0`, then (a) y can not be between 0 and `8/5` (b) y can not be between `-8/5 and 8/5` (c) y cannnot be between `-8/5 and 0`(d) y cannot be between `-16/5 and 0`

Options

A. a can not be between `0 and 8/5`
B. y can not be between `-8/5 and 8/5`
C. you can not be between `-8/5 and 0`
D. y can not be between `-16/5 and 0`

Answer

To solve the problem, we start with the given equation:

\[ x^2 + xy - y^2 + 2x - y + 1 = 0 \]

We will treat this as a quadratic equation in \( x \). A quadratic equation in the form \( ax^2 + bx + c = 0 \) has real solutions if the discriminant \( D \) is greater than or equal to zero.

### Step 1: Identify coefficients
In our equation, we can rewrite it as:

\[ x^2 + (y + 2)x + (-y^2 - y + 1) = 0 \]

Here, we identify:
- \( a = 1 \)
- \( b = y + 2 \)
- \( c = -y^2 - y + 1 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (y + 2)^2 - 4(1)(-y^2 - y + 1) \]

### Step 3: Expand and simplify the discriminant
Expanding \( D \):

\[ D = (y^2 + 4y + 4) - 4(-y^2 - y + 1) \]

\[ D = y^2 + 4y + 4 + 4y^2 + 4y - 4 \]

Combining like terms:

\[ D = 5y^2 + 8y + 0 \]

### Step 4: Set the discriminant greater than or equal to zero
For real solutions, we need:

\[ 5y^2 + 8y \geq 0 \]

### Step 5: Factor the inequality
Factoring out \( y \):

\[ y(5y + 8) \geq 0 \]

### Step 6: Find the critical points
The critical points occur when:

1. \( y = 0 \)
2. \( 5y + 8 = 0 \) which gives \( y = -\frac{8}{5} \)

### Step 7: Test intervals
We will test the intervals determined by the critical points \( y = -\frac{8}{5} \) and \( y = 0 \):

- For \( y < -\frac{8}{5} \) (e.g., \( y = -2 \)):
\[ (-2)(5(-2) + 8) = (-2)(-10 + 8) = (-2)(-2) = 4 \quad (\text{positive}) \]

- For \( -\frac{8}{5} < y < 0 \) (e.g., \( y = -1 \)):
\[ (-1)(5(-1) + 8) = (-1)(-5 + 8) = (-1)(3) = -3 \quad (\text{negative}) \]

- For \( y > 0 \) (e.g., \( y = 1 \)):
\[ (1)(5(1) + 8) = (1)(5 + 8) = (1)(13) = 13 \quad (\text{positive}) \]

### Step 8: Conclusion
The inequality \( y(5y + 8) \geq 0 \) holds for:

- \( y \leq -\frac{8}{5} \) or \( y \geq 0 \)

Thus, \( y \) cannot be between \( -\frac{8}{5} \) and \( 0 \).

### Final Answer
The correct option is (c) \( y \) cannot be between \( -\frac{8}{5} \) and \( 0 \).

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