If x is real then the values of `(x^(2) + 34 x - 71)/(x^(2) + 2x - 7)` does not lie in the interval

Options

A. Lies between 4 and 7
B. Lies between 5 and 9
C. Has no value between 4 and 7
D. Has no value between 5 and 9

Answer

To solve the problem, we need to determine the values of \( y = \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \) that do not lie in a certain interval. We will analyze this expression step by step.

### Step 1: Define the equation
Let
\[ y = \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ y(x^2 + 2x - 7) = x^2 + 34x - 71 \]
This simplifies to:
\[ yx^2 + 2yx - 7y = x^2 + 34x - 71 \]

### Step 3: Rearrange the equation
Rearranging the equation, we get:
\[ yx^2 - x^2 + 2yx - 34x - 7y + 71 = 0 \]
Factoring out \( x^2 \) and \( x \) terms, we have:
\[ (1 - y)x^2 + (2y - 34)x + (71 - 7y) = 0 \]

### Step 4: Apply the condition for real \( x \)
For \( x \) to be real, the discriminant of this quadratic equation must be non-negative:
\[ D = (2y - 34)^2 - 4(1 - y)(71 - 7y) \geq 0 \]

### Step 5: Expand the discriminant
Expanding the discriminant:
\[ D = (2y - 34)^2 - 4(71 - 7y + 7y - 7y^2) \]
\[ = 4y^2 - 136y + 1156 - 4(71 - 7y + 7y^2) \]
\[ = 4y^2 - 136y + 1156 - 284 + 28y - 28y^2 \]
\[ = -24y^2 - 108y + 872 \]

### Step 6: Set the discriminant greater than or equal to zero
Now we need to solve:
\[ -24y^2 - 108y + 872 \geq 0 \]

### Step 7: Divide by -4 to simplify (reversing the inequality)
Dividing by -4 (and reversing the inequality):
\[ 6y^2 + 27y - 218 \leq 0 \]

### Step 8: Find the roots of the quadratic equation
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 6, b = 27, c = -218 \):
\[ D = 27^2 - 4 \cdot 6 \cdot (-218) = 729 + 5248 = 5977 \]
Calculating the roots:
\[ y = \frac{-27 \pm \sqrt{5977}}{12} \]

### Step 9: Determine the intervals
Let the roots be \( y_1 \) and \( y_2 \). The quadratic opens upwards (as the coefficient of \( y^2 \) is positive), so the values of \( y \) that satisfy \( 6y^2 + 27y - 218 \leq 0 \) will lie between the roots \( y_1 \) and \( y_2 \).

### Step 10: Conclusion
The values of \( y \) that do not lie in the interval \( (y_1, y_2) \) are:
\[ (-\infty, y_1] \cup [y_2, \infty) \]

### Final Answer
The values of \( y \) do not lie in the interval \( (y_1, y_2) \).

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